BUT THE ONE WHOSE GENEALOGY IS NOT TRACED FROM THEM COLLECTED A TENTH FROM ABRAHAM: ho de me genealogoumenos (PPPMSN) ex auton dedekatoken (3SRAI) ton abraam: (Heb 7:4; Genesis 14:19,20 )

But (de) introduces a striking contrast.

Genealogy...traced (1075) (genealogeo from genea [from ginomai = to become] = a generation, a race, posterity, that which has been begotten + lego = to choose, to pick out, to reckon) means to reckon or trace a genealogy or ancestry.

The only other use of genealogeo in Scripture is in the Septuagint (LXX)...

1 Chronicles 5:1 Now the sons of Reuben the first-born of Israel (for he was the first-born, but because he defiled his father's bed, his birthright was given to the sons of Joseph the son of Israel; so that he is not enrolled in the genealogy (Hebrew = yachas = to reckon genealogically ; Lxx = genealogeo) according to the birthright.

The one whose genealogy is not traced from them - More literally this reads "he who is not genealogically derived from them". Obviously, this description refers to Melchizedek who was not a descendant of Abraham.

In this verse the writer substantiates the fact that the Melchizedekian priesthood and thus Christ's  priesthood is superior to the Aaronic (Levitical) priesthood because Levi's great-grandfather Abraham paid tithes to Melchizedek and Levi, though as yet unborn, was involved in this transaction (see Hebrews 7:9 note). Obviously Levi was not born until many years after the event in Genesis 14 and yet as the Jews well knew Abraham was the father the Hebrew race. It follows that all twelve tribes including the tribe of Levi (and the priestly family) were represented in Abraham in Genesis 14 when he recognized the superiority of Melchizedek by paying a tenth to him and receiving Melchizedek's priestly blessing. And as he goes on to state in the next verse no one disputes the fact that "the lesser is blessed by the greater". Thus in this dual manner the surpassing greatness of the royal priesthood of Melchizedek is emphasized.

Collected a tenth (1183) (dekatoo from dekatos = tenth) means to take tithes or a tenth from (active voice as in Hebrews 7:6) or to pay tithes (passive voice in Hebrews 7:9). There is one use in the Septuagint (LXX)...

Nehemiah 10:37 We will also bring the first of our dough, our contributions, the fruit of every tree, the new wine and the oil to the priests at the chambers of the house of our God, and the tithe of our ground to the Levites, for the Levites are they who receive the tithes (Hebrew = 'asar = to take or give a tenth; Lxx = dekatoo) in all the rural towns.

The perfect tense signifies that this collection was made historically at a point in time in the past and that it remains on the written record of Scripture.

Vincent explains it this way...

Melchisedec, who has no part in the Levitical genealogy, and therefore no legal right to exact tithes, took tithes from the patriarch himself. Hence he was greater than Abraham. The right of the Levitical priest to receive tithes was only a legal right, conferred by special statute, and therefore implied no intrinsic superiority to his brethren; but Melchisedec, though having no legal right, received tithes from Abraham as a voluntary gift, which implied Abraham’s recognition of his personal greatness.

MacDonald comments that this...

was an unusual and unconventional transaction. Abraham, called to be the father of the nation from which Messiah would come, was paying deference to one who was not connected with the chosen people. Melchizedek’s priesthood leaped over racial barriers. (MacDonald, W & Farstad, A. Believer's Bible Commentary: Thomas Nelson or Logos)

AND BLESSED THE ONE WHO HAD THE PROMISES: kai ton echonta (PAPMSA) tas epaggelias eulogeken (3SRAI): (Hebrews 6:13-15; 11:13,17; Genesis 12:2,13; 13:14-17; 17:4-8; 22:17,18; Acts 3:25; Romans 4:13; 9:4; Galatians 3:16)

Melchizedek collected a tenth from Abraham, and also blessed Abraham, this latter act demonstrating (including Abraham's receipt of the blessing) the implied superiority of Melchizedek.

Blessed the one who had the promises - Melchizedek accepted the exalted position Abraham accorded him, and gave the patriarch his blessing.

Blessed (2127)(eulogeo from eu = good + logos = word) is the verb form meaning to speak well of, to celebrate with praises, to praise. When used by men toward men it means to speak well of with praise and thanksgiving (English "eulogize"). It means to invoke God’s blessing upon them.

Blessed is in the perfect tense indicating that it has occurred at a point in time in the past and still remains on record in Genesis.

Calvin writes that

Let us first know what the word blessed means here. It means indeed a solemn praying, by which he who is invested with some high and public honour, recommends to God men in private stations and under his ministry. Another way of blessing is when we pray for one another, which is commonly done by all the godly. But this blessing mentioned by the apostle was a symbol of greater authority. Thus Isaac blessed his son Jacob, and Jacob himself blessed his grandsons (Gen. 27:27, 48:15). This was not done mutually, for the son could not do like the father; but a higher authority was required for such a blessing as this. And this appears more evident still from Num. 6:23, where a command is given to the priest to bless the people, and then a promise is immediately added, that they would be blessed whom they blessed. It hence appears that the blessing of the priest depended on this,—that it was not so much man’s blessing as that of God. For as the priest in offering sacrifices represented Christ, so in blessing the people he was nothing more than a minister and legate of the supreme God

Adam Clarke quotes Macknight...

The blessing here spoken of...is not the simple wishing of good to others, which may be done by inferiors to superiors; but it is the action of a person authorized to declare God’s intention to bestow good things on another.

Leon Morris explains that...

There are senses of the word "bless" in which men "bless" God, i.e., praise him, or in which an inferior prays that God will prosper some superior. But the word is not used in such a way here. It is rather the official pronouncement given by an authorized person. When that happens, there is no denying that it proceeds from a superior. (Gaebelein, F, Editor: Expositor's Bible Commentary 6-Volume New Testament. Zondervan Publishing)

Who had the promises - This descriptive phrase is used as a "synonym" for Abraham ("the possessor of the promises") who had received the promises. The writer by using this description exalts Abraham in order still more to exalt Melchizedek.

The promises include those described in the following passages (these Scriptures are not an exhaustive record of the promises to Abraham, eg, Psalm 105:42, Acts 7:17, Romans 4:13ff, Hebrews 119, 17, et al)...

Genesis 12:1 Now the LORD said to Abram, "Go forth from your country, And from your relatives And from your father's house, To the land which I will show you; 2 And I will make you a great nation, And I will bless you, And make your name great; And so you shall be a blessing; 3 And I will bless those who bless you, And the one who curses you I will curse. And in you all the families of the earth shall be blessed."...7 And the LORD appeared to Abram and said, "To your descendants I will give this land." So he built an altar there to the LORD who had appeared to him.

Genesis 13:15 for all the land which you see, I will give it to you and to your descendants forever. 16 "And I will make your descendants as the dust of the earth; so that if anyone can number the dust of the earth, then your descendants can also be numbered. 17 "Arise, walk about the land through its length and breadth; for I will give it to you."

Genesis 15:18 On that day the LORD made a covenant with Abram, saying, "To your descendants I have given this land, From the river of Egypt as far as the great river, the river Euphrates

Genesis 17:4 "As for Me, behold, My covenant is with you, And you shall be the father of a multitude of nations. 5 "No longer shall your name be called Abram, But your name shall be Abraham; For I will make you the father of a multitude of nations. 6 "And I will make you exceedingly fruitful, and I will make nations of you, and kings shall come forth from you. 7 "And I will establish My covenant between Me and you and your descendants after you throughout their generations for an everlasting covenant, to be God to you and to your descendants after you. 8"And I will give to you and to your descendants after you, the land of your sojournings, all the land of Canaan, for an everlasting possession; and I will be their God."

Genesis 22:17 indeed I will greatly bless you, and I will greatly multiply your seed as the stars of the heavens, and as the sand which is on the seashore; and your seed shall possess the gate of their enemies.

Acts 3:25 "It is you who are the sons of the prophets, and of the covenant which God made with your fathers, saying to Abraham, 'AND IN YOUR SEED ALL THE FAMILIES OF THE EARTH SHALL BE BLESSED.'

Galatians 3:16 Now the promises were spoken to Abraham and to his seed. He does not say, "And to seeds," as referring to many, but rather to one, "And to your seed," that is, Christ.

Hebrews 6:13 For when God made the promise to Abraham, since He could swear by no one greater, He swore by Himself, 14 saying, "I WILL SURELY BLESS YOU, AND I WILL SURELY MULTIPLY YOU." 15 And thus, having patiently waited, he obtained the promise.

Normally one would suppose that the one who had the promises (Abraham) might be considered to have been above being blessed by any other man. However, as the writer has made amply clear, Melchizedek is not "any other man" but is indeed a unique and great king-priest. This fact again accentuates that the Priesthood of Jesus according to the Order of Melchizedek was greater than the Levitical priesthood. This may seem repetitious to us today for the Temple in Jerusalem has long disappeared and most of us are "wild olive branches" (Gentiles) and not likely to seek to cling to the old ways of the Levitical priesthood. But for a Jew in the time of this writing (the Temple was still standing), this paradigm shift called for such inspired repetition.

Promises (1860) (epaggelia from epí = intensifies verbal meaning + aggéllo = to tell, declare) originally referred to an announcement or declaration (especially of a favorable message) but in later Greek came to mean a declaration to do something with the implication of obligation to carry out what is stated (thus a promise or pledge). Epaggelia was primarily a legal term denoting summons, a promise to do or give something (or to refrain from doing something). The NT promises refer almost exclusively the promises of God. For more on promise see ISBE article.

Clearly promise is a key word in this epistle where epaggelia is used 14 times in 13 verses (27.4% of the 51 NT uses) (See notes Hebrews 4:1, Hebrews 6:12, 6:15, 6:17, Hebrews 7:6, Hebrews 8:6, Hebrews 9:15, Hebrews 10:36, Hebrews 11:9, 11:13, 11:17, 11:33, 11:39)

Expositor's Greek Testament writes that...

The point which the writer here brings out is that although Abraham had the promises, and was therefore himself a fountain of blessing to mankind and the person on whom all succeeding generations depended for blessing (cp Ge 12:3 "And I will bless those who bless you, and the one who curses you I will curse. And in you all the families of the earth shall be blessed."), yet Melchizedek blessed him!

Vincent explains that...

BUT WITHOUT ANY DISPUTE THE LESSER IS BLESSED BY THE GREATER: choris e pases antilogias to elatton hupo tou kreittonos eulogeitai (3SPPI): (1 Timothy 3:16 ) (Hebrews 11:20,21; Genesis 27:20-40; 28:1-4; 47:7-10; 48:15-20; 49:28; Numbers 6:23-27; Deuteronomy 32:1; 2 Samuel 6:20; 1 Kings 8:55; 2 Chronicles 30:27; Luke 24:50,51; 2 Corinthians 13:14)

This verse represents a brief "parenthesis" in which the writer states a generalization that everyone recognizes.

Vincent states the writer is...

Asserting a principle which no one thinks of questioning: it is the less who is blessed, and the greater who blesses.

The International Children's Bible paraphrase accurately conveys the sense of this verse rendering it...

And everyone knows that the more important person blesses the less important person.

The NLT accurately paraphrases it...

And without question, the person who has the power to bless is always greater than the person who is blessed.

Without (5565) (choris) as a preposition (its more frequent use) means “apart from, without, separate from.

Dispute (485) (antilogia from antilego = contradict <> anti = against + lego =  speak) literally means "a word spoken against" or speaking against and so talking back, face to face, in opposition against (a dispute involving opposite opinions). Our English word dispute describes a disagreement, a quarrel, an argument or a verbal controversy. Antilogia describes contradiction or controversy with the added sense that strife is involved. Strife means bitter sometimes violent conflict or dissension.

The lesser - In the present context this alludes to Abraham.

Lesser (1640) (elasson) is the comparative degree of mikros meaning “little”and denotes “less” in quality, age or rank, this latter meaning applying to the present context.

The greater -  In the present context this alludes to Melchizedek.

Greater (2909) (kreisson from kratos = strong, which denotes power in activity and effect) serves as the comparative degree of agathos, “good” (good or fair, intrinsically). Kreisson is a key word in Hebrews (click the 12 uses in Hebrews out of 19 NT uses).

The point is that it is such a universally held truth that the inferior individual is blessed by the superior one, there is simply no room for dispute about this matter. This same saying was common in the Greek secular writings when a writer wished to stress the certainty of what he said. This does not signify any personal or moral inferiority, of course, but simply an inferiority of position.

Although this principle was known in secular Greek, as Fausset comments from a Scriptural viewpoint...

The principle that the blesser is superior to him whom he blesses, holds good only in a blessing given with divine authority; not merely a prayerful wish, but one that is divinely efficient in working its purport, as that of the patriarchs on their children: so Christ’s blessing, Lu 24:51 ("while He was blessing them, He departed from them"); Acts 3:26 ("For you first [Jews] God raised up His servant and sent Him to bless you [how?] by turning every one of you from your wicked ways").

As the blesser, Melchizedek indisputably was superior to Abraham. If Melchizedek was superior to Abraham, then he must also be superior to the Levites, Abraham’s descendants. Consequently, his priesthood is superior to theirs.

John Brown offers a qualifying comment...

And without all contradiction the less is blessed of the better” (v. 7). This verse summarizes the argument contained in vv. 4–6. These words are plainly to be understood with limitations. It does not follow that, because a priest under the law blessed the king, be was in a civil capacity the king’s superior, any more than that a Christian minister instructing or even reproving a man of high civil rank who is a member of the church of which he is pastor, is civilly his superior. The apostle’s argument is: The person who accepts of priestly benediction from an individual acknowledges his spiritual superiority, just as the highest authority in the land, if he were becoming a member of a voluntary Christian society, would acknowledge that its pastor was ‘over him in the Lord’

Leon Morris explains that...

In the Genesis account Melchizedek makes no claims nor does Abraham concede anything in words. But the patriarch gave up a tenth of the spoils, thus implicitly acknowledging the superior place of Melchizedek. And Melchizedek proceeded to bless Abraham, accepting the implied superiority. The situation is clear to all parties. There is no need to spell it out. And the author is simply drawing attention to what the narrative clearly implies when he brings out the superior status of Melchizedek. Even when Abraham is seen as the one "who had the promises," Melchizedek is superior.  (Gaebelein, F, Editor: Expositor's Bible Commentary 6-Volume New Testament. Zondervan Publishing)

Abraham was blessed by Melchizedek. In a similar fashion Aaron was to bless the people in Numbers 6 

23 "Speak to Aaron and to his sons, saying, 'Thus you shall bless the sons of Israel. You shall say to them:
24 The LORD bless you, and keep you;
25 The LORD make His face shine on you, And be gracious to you;
26 The LORD lift up His countenance on you, And give you peace.'
27 "So they shall invoke My name on the sons of Israel, and I then will bless them."

Most of us are Gentile believers but it behooves as we read these arguments to try to place ourselves in the sandals of these first century Jewish readers who had always revered Abraham as their ancestral father and national hero. And yet now they learn the their father Abraham actually acknowledged a "non-Jewish" priest as his superior! Imagine how they must have felt, especially in light of the fact that this very information had been recorded by Moses and they had failed to recognize or understand its significance!

Pentecost explains that...

The Oriental view of superiority and subordination is a concept foreign to our modern Western culture. Therefore it is popular among unbelievers to stand in judgment of God and raise questions like, “How can a loving God allow suffering?” or “How could a God of love send people to hell?” Sometimes this attitude carries over into our Christian lives after we trust Christ as Savior, and we find ourselves questioning His wisdom, His love, or His understanding. interestingly, the Bible seems to indicate that real understanding of God’s work in our lives begins with our humble acknowledgement that He is superior to us, and based on that alone—whether we understand or not—we submit to His will in our lives. This was the lesson Job had to learn, and it was Abraham’s attitude as well. Shouldn’t it be ours? (Pentecost, J. D., & Durham, K. Faith that Endures: A Practical Commentary on the Book of Hebrews. Grand Rapids, MI: Kregel Publications)

Andrew Murray in his devotional commentary "Holiest of All" has the following points of application relating to this section...

1. Melchizedek blessed Abraham. The work of thy High Priest, O my soul, Is simply blessing. Learn to think this of Jesus, and seek to have a great confidence that He delights to bless. He Is nothing but a fountain of blessing; rejoice greatly In this and trust Him for it.

2. Remember that the all-comprehensive blessing of thy Melchizedek in heaven is—the Holy Spirit from heaven in thy heart. As It Is written: "Christ redeemed us, that the blessing of Abraham might come upon us, in Jesus Christ; that we might receive the promise of the Spirit through faith." The Holy Spirit "abiding continually" in the heart is the high-priestly blessing.

3. This day He comes to meet thee, as thou returnest from the battle weary and faint. Bow before Him, and let Him bless thee I "Even as the Holy Ghost saith, To-day." Believe that Jesus is all to thee.