Hebrews 7:24 refutes the LDS claim to have a Melchizedek Priesthood held by millions of priests (that is, every worthy LDS male age 19 and up). When it says that Jesus has an “unchangeable priesthood,” the word “unchangeable” is actually the Greek a-parabaton, which means literally “non-transferable.” Jesus is the unique holder of that office. His office cannot be passed to anyone else. He is the one and only Melchizedek priest.
To back that up, let’s look at the rest of what the author of Hebrews says about the “order of Melchizedek.” The Greek term taxis (pronounced “tax-iss”) used in this phrase is used only three times in the New Testament outside of this letter, all in the sense of sequential order. The most famous is 1 Corinthians 14:40: “Let all things be done decently and in order.” In Colossians 2:5, Paul tells the church that he rejoices to see their “order” and the steadfastness of their faith in Christ. In Luke 1:8, Zechariah “executed the priest’s office in the order of his course” (New King James “in the order of his division,” that is, when his unit’s turn came to serve on duty).
The references to the “order of Melchizedek” in the book of Hebrews are all based on the Greek translation of Psalm 110:4. Here the word taxis is best translated “arrangement” or “classification” (as in our English term “taxonomy,” borrowed from this word), because all of the other clues in this letter point to a “classification” or category of priest into which only one member fits. Hebrews 7:11 contrasts the Melchizedek “kind” (taxis) of priest with the Aaron “kind” (same word).
Hebrews 7:3 observes that the Melchizedek of Genesis 14 appears to be “made like the Son of God.” He appears with no mention of ancestors or descendants. His chief qualifications are that he has “neither a beginning of days nor end of life,” and “remains a priest forever.” Some have suggested that this mysterious character is actually a pre-incarnational appearance of Christ himself. While this possibility is attractive, it would require him to be able to bring bread and wine to Abraham, and receive tithes from Abraham, both of which would seem to require a body, which he does not receive until his conception almost 2000 years later. (And 9:27 rules out re-incarnation.)
Melchizedek is presented in Hebrews as a Lone Ranger, with no established priesthood to which he belongs; Jesus is the same kind of priest. Jesus gets his priesthood, not by DNA or by human decree; he gets it from “the power of an indestructible life” (7:15), which is his “likeness” to Melchizedek, whose priesthood continues “forever.” We are told that Jesus guarantees a better covenant than the other kind of priests, because they were prevented by death from continuing in office, whereas because he continues forever, he has an “un-transferable” priesthood – it cannot and need not be passed on to anyone else (7:22-24). Because he always lives, Jesus is always able to plead the case of sinners who draw near to God (7:25).
So the entire argument in the book of Hebrews is that Jesus’ priesthood puts him in a class by himself. No one else is qualified to permanently take away sins. The other kind of priest couldn’t do it. Priesthood is all about atonement for sin, not the authority to act for God in any other way. And because of Jesus’ once for all sacrifice for sin, there is no longer any need for priesthood of any kind (Hebrews 10:11-18), except the one who always lives to plead our case with God.
The only way that Christians today can speak of priesthood is in a less-than-literal sense. When Christian denominations (such as Catholic, Orthodox, and Episcopal) speak of priests, they are speaking of leaders, pastors, shepherds of souls, not people who atone for sin. And the apostle Peter proclaims what Luther identified as the priesthood of all believers (“you are a royal priesthood,” 1 Peter 2:9), a priesthood that includes both women and men, trained and untrained. Because we now have direct access to God (Hebrews 10:19-22), all who trust in the one and only Melchizedek priest for their salvation have authority to speak and act for God, as long as we speak and do only what God has clearly taught in God’s word.
So the Bible gives us no evidence for a Melchizedek priesthood with thousands of priests who have authority to speak and act for God. When you as a historic Christian get asked, “Where do you get your authority?”, you can say, “From Jesus. You have the Melchizedek priesthood. I have the one and only Melchizedek priest.”