Do modern Jews believe in a priesthood after the order of Melchisedec?

Question

Do modern Jews believe in a priesthood after the order of Melchisedec? Is there anything taught about it? From the Christian book of Hebrews 7:11:

...what further need was there that another priest should rise after the order of Melchisedec...

It appears that Paul is speaking about a concept that was familiar to Jews at the time. Was there a Judaic group of priests "after the order of Melchisedec?" Or was Paul introducing a Christian concept the ancient Jews would not have recognized from their own tradition?

Edit: I would like to extend my profound thanks and gratitude for everyone's patience and efforts to address my question. I knew it would create some discomfort, but hoped the discussion would prevail as I have at no time meant to dishonor or disparage Judiasm in any way. It should come to no surprise that I am Christian, and in my own studies I intentionally force myself to step back and try to see a larger picture than the one bequethed to me by my parents. The links to references (even the words you have used) have provided me a rich trove of knowledge through which I may continue my studies. A mere scratch on the surface to you, I am sure, but an entire landscape to study for me.

Answer 1

Malkitezedek was a priest contemporary with Avram. He lost or ceded his special status to Avram (Tz'enah Ur'enah and R' Daniel Raccah cited here).

Avram's (i.e. Avraham's) grandson Yaakov had twelve sons, one of which was Levi. When Israel received the torah a few generations later, the tribe of Levi was set aside for service to God, and within the tribe of Levi, Aharon and his the descendants were designated exclusively as priests (kohanim). The kohanim are the only legitimate priests in Judaism since the giving of the torah. Anybody descended from or seeking to emulate Malkitzedek, who is not also a kohein, has no priestly status in Judaism.

You asked how Paul's message would have been understood by Jews of his time. Jews who read the entire verse you cited would have rejected the heresy wholesale. I've bolded the parts that are especially problematic:

If perfection could have been attained through the Levitical priesthood—and indeed the law given to the people established that priesthood—why was there still need for another priest to come, one in the order of Melchizedek, not in the order of Aaron? (source)

Perfection -- if, by that, Paul means doing God's will and fulfilling our purpose -- is attained through the law and its proper observance. There was no "need" for a new priest. And a "priest" explicitly not descended from Aharon is not a priest. In Paul's time Jews had already been keeping the torah for more than a thousand years; they knew this and would not have been swayed by his assertions that the torah could be superseded.

Early Christianity, on the other hand, needed to explain how their non-kohein founder was salvivic, and there were other benefits to delegitimizing or superseding both torah and the priesthood.

Answer 2

The reference in Psalms Tehillim 110:4

נִשְׁבַּע יְהֹוָה | וְלֹא יִנָּחֵם אַתָּה כֹהֵן לְעוֹלָם עַל דִּבְרָתִי מַלְכִּי צֶדֶק:

The Lord swore and will not repent; you are a priest forever because of the speech of Malchizedek.

Rashi explains in 110:1 that this refers to Avraham.

The word of the Lord to my master: Our Rabbis interpreted it as referring to Abraham our father, and I shall explain it according to their words (Mid. Ps. 110:1): The word of the Lord to Abraham, whom the world called “my master,” as it is written (Gen. 23: 6): “Hearken to us, my master.”

As a result, just as Shem son of Noach was king of Shalem and a priest of Hashem, so to would Avraham be the progenitor of the Kings and Priests of Hashem's people. The Kings would be from David Hamelech and the Priests from Aharon.

Rashi

you are a priest forever because of the speech of Malchizedek: From you will emerge the priesthood and the kingship that your children will inherit from Shem your progenitor, the priesthood and the kingship, which were given to him. דִבְרָתִי מלכי-צדק. The “yud” is superfluous, like (Lam. 1: 1): “the city that was once so populous (רבתי).” Because of the speech of Malchizedek, because of the command of Malchizedek. You are a priest, Heb. כהן. The word כהן bears the connotation of priesthood and rulership, as (II Sam. 8:18): “and David’s sons were chief officers.”

As it says in II Sam. 8:18

וּבְנָיָהוּ בֶּן יְהוֹיָדָע וְהַכְּרֵתִי וְהַפְּלֵתִי וּבְנֵי דָוִד כֹּהֲנִים הָיוּ:

And Benayahu the son of Yehoyada [was over] the archers and the slingers; and David's sons were chief officers.

Answer 3

The other answer here has the basics on the Melchizedek in Genesis and the Kohanim as the only legitimate Jewish priests.

Chapter 7 of Hebrews is an attempt by "Paul"(authorship of this particular letter by him has been disputed by some of the earliest church commentators, Eusebius, for example) to try and prove that Yeshu was the new, permanent High Priest, even though he was from the Tribe of Judah. I'm not going to go over it line by line, the whole chapter is here, but the basic idea is that "Paul" writes that Melchizedek "resembles the Son of God", then tries to show that by Abraham's tithing to Melchizedek, and the quote of Psalm 110:4:

"The Lord swore and will not repent, you are a priest forever after the manner(JPS)/order(most non-Jewish translations) of Melchizedek"

that(as is usual in their supercessionist theology)the "Old Covenant", and the Levitical Priesthood, have been replaced by the "New Covenant", and that Yeshu is the new, permanent high priest, who offered himself as sacrifice for everyone's sins, etc.

At Qumran in cave 11 they found a badly worn text 11QMelchizedek, in which he seems to have a priestly/Messianic role(hard to tell for certain from the condition of the manuscript), so there were extra-Biblical legends floating around about him in Late Second Temple times. Some Qumran texts also mention his direct opposite, Melchiresha. Melchizedek also appears a few centuries later on as the subject of one of the Gnostic Nag Hammadi tracts.

Answer 4

Wikipedia has some interesting information about this subject (lightly edited):

The Law of Moses stipulates that only the male descendants of Aaron be commissioned to serve as Jewish priests before the God of Israel and the Jewish nation. This commission is in Judaism "a covenant of everlasting priesthood" ("berit kehunat olam," Num. 25:13) and not eligible for replacement by other tribes of Israel... Although the Book of Genesis affirms that Melchizedek was "priest of God Most High" (Gen. 14:18), The Midrash and Babylonian Talmud maintain that the priesthood held by Melchizedek, who pre-dated Aaron by six generations (Abraham, Isaac, Jacob, Levi, Kehoth, Amram, Aaron), was given in his stead to Abraham who in turn passed it on to his descendants... Midrashic literature attributes this transition as a consequence due to Melchizedek preceding the name of Abraham to that of God, such as in the Midrash Rabbah to Genesis and Tractate Nedarim (32b), while some Jewish commentators, such as Chaim ibn Attar, write that Melchizedek gave the priesthood to Abraham willingly.