According to the writer of Hebrews, the law which established the Levitical priesthood to offer sacrifices on behalf of the sins of the people, has been annulled. ~ Hebrews 7:16-19; 9:10

The author of Hebrews expounds on the ministry of Jesus as High Priest in contrast to the priests who served under the law. It is important to keep in mind that at the time of the writing of the book of Hebrews, the Temple was still standing and sacrifices were still being offered, even though they had been rendered obsolete by the sacrifice of Jesus.

For this reason, the writer of Hebrews says to his audience, if Jesus were on earth he would not be a priest because those priests (who serve in the earthly temple) offer gifts and sacrifices according to the Law. ~ 8:4

Notice closely the language used in Hebrews 8:4, If Jesus were on earth, he would not be a priest. Jesus’s priesthood belongs in the Heavens where God’s Throne is. Jesus, and Jesus alone is the only one who could serve as High Priest in the very presence of God in Heaven above.

Jesus is overqualified to be a priest on earth! This is the fundamental difference that the writer of Hebrews is teaching us regarding the Levitical priesthood, and the Priesthood of Jesus, after the Order of Melchisdek. The priests under the law could only serve on earth where men die, but Christ’s Priesthood transcends that which was carried out on earth.

Because his ministry has surpassed the ministry of the old priesthood, there will never be the need for their service, ever again. Jesus has opened for us a new and living way (10:19-22) and serves as priest forever and mediator of an everlasting covenant established by his blood. 

Those who served as priests under the law, could never please God because their work was never finished. The priests were limited by their own imperfections and did not continue forever because they were subject to death as fallen humanity. Jesus however lives forever in resurrection power.

Now there have been many of those priests, since death prevented them from continuing in office; but because Jesus lives forever, he has a permanent priesthood. Therefore he is able to save completely those who come to God through him, because he always lives to intercede for them. Such a high priest truly meets our need—one who is holy, blameless, pure, set apart from sinners, exalted above the heavens. Unlike the other high priests, he does not need to offer sacrifices day after day, first for his own sins, and then for the sins of the people. He sacrificed for their sins once for all when he offered himself. For the law appoints as high priests men in all their weakness; but the oath, which came after the law, appointed the Son, who has been made perfect forever. ~ Hebrews 7:23-28NIV