First of all it's a natural implication of Believers now being God's Temple/Tabernacle. A doctrine I've defended mostly on my Prophecy Blog, in posts like the one on there not being a Millennial Temple, or on The Church being The Bride of Christ. Note, the latter of those posts was made back when I was more Dispensational then I am now, I have revised my view on that subject since.
But as far as passages specifically about the Priesthood of Believers. We could start with 1 Peter 2:4-10.
"To whom coming, as unto a living stone, disallowed indeed of men, but chosen of God, and precious, Ye also, as lively stones, are built up a spiritual house, an holy priesthood, to offer up spiritual sacrifices, acceptable to God by Jesus Christ. Wherefore also it is contained in the scripture, Behold, I lay in Sion a chief corner stone, elect, precious: and he that believeth on him shall not be confounded. Unto you therefore which believe he is precious: but unto them which be disobedient, the stone which the builders disallowed, the same is made the head of the corner, and a stone of stumbling, and a rock of offence, even to them which stumble at the word, being disobedient: whereunto also they were appointed. But ye are a chosen generation, a royal priesthood, an holy nation, a peculiar people; that ye should show forth the praises of him who hath called you out of darkness into his marvellous light: Which in time past were not a people, but are now the people of God: which had not obtained mercy, but now have obtained mercy."In The Book of Revelation it's clearly taught in Chapter 1 verses 4-6. And it becomes a part of determining who the 24 Elders are because of Chapter 5 verses 6-10. (I view the 24 Elders as among those saints who were Resurrected in 30 AD in Matthew 27:51-53.)
As far as those who might object to this because of how Torah centric they are, and would thus be offended by the notion of the Aaronic Levitical Priesthood becoming obsolete. This is anticipated in The Torah itself in Exodus 19:5-6. In fact I think the above quoted 1 Peter passage was drawing on these verses.
"Now therefore, if ye will obey my voice indeed, and keep my covenant, then ye shall be a peculiar treasure unto me above all people: for all the earth is mine: And ye shall be unto me a kingdom of priests, and an holy nation. These are the words which thou shalt speak unto the children of Israel."This wasn't fulfilled during the Old Testament because Israel kept disobeying. Now with the Blood of Jesus shed, the true intent of Yahuah can finally be fulfilled.
The word translated "Everlasting" or "Forever' or "Eternal" when referring to things like the Levitical Priesthood is Olam, which means age or eon, it does not actually mean forever. Whether it's Exodus 40:15, or Leviticus 16:34, or Leviticus 24:8, or Numbers 25:13. Same with Exo 21:6, Exo 27:21, Exo 28:43, Exo 29:28, Lev 6:18, Lev 6:22.
In Deuteronomy 33:27, Olam is used of the "everlasting arms" but a different word is used to call God Eternal.
Likewise the phrase "all the days", which is introduced about time periods that have an end in Genesis 3:14-17. And again in Genesis 5. And it's also used of the Nazarite vow in Numbers 6. If "all the days" is being used of something that is also defined as an Olam, an Age, then it clearly means all the days of that age, just as it can also mean all the days of someone's life. Taking the phrase to inherently mean all the days of eternity, it completely illogical.
The Book of Hebrews is considered the definitive discussion of this doctrine. Before mentioning it I wanted to show that the Doctrine isn't dependent on Paul, since some people want to reject Paul.
Much of that is built around Melchizedek, and his Priesthood as superior to Aaron's. Psalm 110 is one of the most quoted Psalms in the New Testament, Jesus clearly defined it's subject as The Messiah when he quotes it in Matthew 22. And it refers to the Messiah as a Priest after the order of Melchizedek.
I also view the Melchizedek narrative of Genesis 14 with it's Bread and Wine as the true Torah precedent for the Last Supper/Eucharist, which was NOT the Passover Seder, that would have been eaten the night after Jesus died on The Cross as the Lambs were slaughtered.
Another Priest who predated the Priesthood of Aaron was Jethro, the Priest of Midian. Jethro was around when the instructions in Exodus, Leviticus and even early Numbers were given. He returns to his own people when the Israeites head to Kadesh, no where is he told that his Priesthood is now over because of Aaron's being founded, he was still the Priest of Midian.
As far as Ezekiel 40-48 goes, that is discussed when I address the Temple doctrine on my Prophecy blog. I don't believe there will be a future Temple building. Regardless the land allotted to the Priests and Levites in Ezekiel I think will probably indeed be given to the descendants of Levi and Aaron in the Millennium and/or New Creation. That will be their post-Resurrection retirement. Likewise with the land given to the Nasi/Prince (David) and his children.
I also agree with those who've argued that Jesus had Aaronic blood via his mother. I think much you can read as eschatological references to Priests or Levites is being not strictly literal. It's Aaron's Blood being passed on to us via Jesus Blood that was shed on the Cross, and us being made His Seed.