Isn't this passage in Psalms 110:4 referring to the Messiah, as being the one who would be a priest forever after the order of Melchizedek?
No, it's not. I don't see the word Messiah anywhere in the chapter.
And the Messiah (the righteous Branch) is also to be the king who will sit on the throne of David.
The Messiah is a descendant of David who will sit on David's throne.
As per verses such as Psalms 110:2
I don't see anything in this verse that has references the Messiah.
and Jeremiah 33:14-18 (which mentions never lacking a man as king, nor a man in the priests position) ,
No, it says that there will be lacking someone who could
potentially perform these jobs. In other words, their lineages would never end, there will always be male descendants of Levi, Aaron and David to fulfill the functions of the Levite, Priest and king.
This verse speaks about the descendant of David who will be a king.
Zechariah 6:12-13 (the Branch will be both king and priest)
No, it says that the shoot of David will build the Temple and rule on his throne and there will also be a priest who will sit on his (ie. the priest's) throne (as the High Priest). And there will be peace between the king and the priest.
So ultimately my question is:
Since the king is to be a descendant of David, of the tribe of Judah, while the priesthood is limited to someone of the tribe of Levi. How can the Messiah be both king and priest, as the scripture lets us know he will be? (Unless you believe there was a change in the law.)
There is no change to the Law, you're just reading into the chapter things that are not there. No where does the chapter mention or reference the Messiah. Jewish sources understand it as talking about either Abraham or David himself.
The fact that it says the word "priest" doesn't mean that it's talking about the Aaronic priests. See 2 Sam. 8:18 where David's sons are called "priests". A "priest" just means some who serves in an officiating position.
The word that you are translating "order", I believe you are misunderstanding as meaning "class, rank" when in that translation it's coming from the root
dvr meaning "speech". So for that translation, "order" means "command".
The word "Melchi-tzedek" doesn't need to be a proper noun, it can also be an adjective meaning: righteous king.
So in conclusion, no, this is not a Messianic prophecy. Obviously a male descendant of Aaron can't also be a male descendant of David, so it's impossible for a priest of Aaronic lineage to also be a kind of Davidic lineage. The Hasmoneans did make themselves kings, however they were punished as all their descendants were killed. As for this chapter of Pslams, i can be understood when translating it in one way to be talking about someone who was both a king and priest, in which case we understand it is referring to Abraham who took the priesthood from Melchitzedek and passed it to his own descendants. If you translate it another way, then it's not talking about someone who is a priest at all, in which case it can be understood to be talking about David, the righteous king who would forever serve G-d as the righteous king.