The purpose of the writer of the Epistle to the Hebrews was to demonstrate the superiority of Christ over the accoutrements of the Mosaic covenant. In chapters 7 and 8, he examines the priestly office of Christ, and compares it to the Mosaic, levitical priesthood. (Just as an aside, this is an internal proof that the epistle must have been written before 70AD, when the destruction of the Temple made that priesthood moot.)
The ways in which Christ's priesthood is superior include His eternality (Heb. 7:3, 17, 24, 28): "He holds His priesthood permanently, because He continues forever." This is contrasted with the string of human priests, due to their mortality (Heb. 7:23). Another way in which He is superior is due to the sufficiency and effectuality of His one-time sacrifice (Heb. 7:27): "He has no need, like those high priests, to offer sacrifices daily..., since He did
this once for all when He offered up Himself."
This passage addresses two errors of the Church of Rome; first, that her clergy have a priesthood; and second, that Christ's sacrifice is repeated in the Mass. Both of those heresies undermine both the sufficiency and eternality of the priestly office of Christ, contrary to both the words and the purpose of the Epistle. That is, if Christ as priest is both eternal and sufficient in that office, there is neither need nor allowance for any other priest or sacrifice. That is why a Protestant church has a minister, not a priest, a man who points believers to their only and all-sufficient hope in Jesus Christ, not in any mere man, no matter how glamorous his robes, titles, and claims may be.
Because of the explicit words of Scripture, I commend the words of the Westminster Larger Catechism (Question 44): "Christ executeth the office of a priest in His once offering Himself as a sacrifice without spot to God, to be a reconciliation for the sins of His people, and in making continual intercession for them."