Holy Orders and the Church
Hebrews 8:5 reveals that the priesthood and Tabernacle of the Old Covenant were shadows of heavenly things. Concerning the Church and the New Covenant, Paul writes of Christ as the High Priest who is now in Heaven (Hebrews 4:14). Because He is alive and continues to hold this office until his return there will never be another 'high priest' in the Church on earth. Just as the Old Covenant religion had priests and Levites, the New Covenant Church has bishops/presbyters and deacons (see 1 & 2 Timothy and Titus).

Christ had appointed Twelve Apostles as well as the Seventy during His earthly ministry to fill the same role as Old Testament priests. The Church consecrates bishops and priests (also referred to elders or presbyters [Titus 1:5; 1 Timothy 4:14]) through the sacrament of Holy Orders. Even in the short time frame in which the New Testament was written the structure of the Church can be clearly seen, so, it was not a neo-pagan invention of a later century as some have charged.

The most important role of the priesthood is dispensing the sacraments.
~ Acts 19:4-6 provides a clear example of Paul baptizing a new convert and then laying hands on him to receive the Holy Spirit (just as baptism and christmation are performed today).

~ Christ gave His power to forgive sins to the Apostles (John 20:22-23) who in turn passed it to their successor the bishops and priest/elders.

~ James 5:13-16 show the sacraments of Holy Anointing and Confession as a work of the elders.

~ Titus 1:5 records Paul's instructions for ordaining new priests/elders by the new bishop, Titus.

~ 1 Corinthians 11:23-30 shows that the consecration of the Eucharist, transforming it from the bread and wine to the actual Body and Blood of Christ though the Holy Spirit , is the priest's most vital and frequent sacramental service for the Church.

~ The Church's role grew in marriage so that by the end of the 2nd century Tertullian writes:
Whence are we to find (words) enough fully to tell the happiness of that marriage which the Church cements, and the oblation confirms, and the benediction signs and seals; (which) angels carry back the news of (to heaven), (which) the Father holds for ratified? [To My Wife Book 2, Chapter 8].

In exhorting elders the Apostle Peter, just as St. Paul, identifies the priesthood as a distinct class of leadership within the Church with specific duties and responsibilities [1 Peter 5:1-4]. Just as I read in Scripture hat Christ assumed the role of High Priest in the New Covenant, I see no verse in the New Testament that abolishes the role of priests but rather redirects it to the Church.

For verily I say unto you, Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law, till all be fulfilled.
Matthew 5:18

Since then these things are manifest to us, and we have looked into the depths of the divine knowledge, we ought to do in order all things which the Master commanded us to perform at appointed times. He commanded us to celebrate sacrifices and services, and that it should not be thoughtlessly or disorderly, but at fixed times and hours. He has Himself fixed by His supreme will the places and persons whom He desires for these celebrations, in order that all things may be done piously according to His good pleasure, and be acceptable to His will. So then those who offer their oblations at the appointed seasons are acceptable and blessed, but they follow the laws of the Master and do not sin. For to the high priest his proper ministrations are allotted, and to the priests the proper place has been appointed, and on Levites their proper services have been imposed. The layman is bound by the ordinances for the laity. [St. Clement, Bishop of Rome, AD 80]



Additional Reference

Holy Orders