Technically you can, but it sounds weird. Technically it changes the meaning a little bit.

Yes, you can. To me, the meaning of "teacher desk" seems very clear, even though it is not a term I have heard much. But then, I have not been in a classroom for decades so whether I have heard it or not means little. I interpret it to mean a desk that is intended to be used by a teacher. It can be the same as all the other desks, but it is reserved for a teacher.

The other reason that ‘teacher’ doesn’t work as an adjective here is because the only thing that makes a desk a teacher’s desk is that it belongs to a teacher (possessive). It isn’t different from any other desk. On the other hand, you can say any of a teacher’s salary a teaching salary a teacher salary And it will mean something. Here ‘teacher’ is still my least favorite of the three, but it doesn’t sound too wrong

(I'm a US speaker). Don't do it. I think it "sounds wrong." I've never heard it. You will be understood, and you OFTEN can use a noun as an adjective that way, but here it sounds very strange. I'm afraid it's just a question of customary usage, and it isn't customary for the word "teacher." Sorry to give such a vague answer. Language is changing in the United States! I have a card in my pocket that says "Driver's license," the possessive form of the word "driver." In some US states, that card says "Driver's license," but in others it says "Driver license.". There is an unmistakable trend toward using apostrophes less and less. But this is a slow process and it seems to be happening with some words and not others. You just have to check usage for each word! "Driver" has made the jump to being usable as an adjective, "teacher" hasn't.

It's just something that's not said. You cannot change "a teaher's desk" with "a teacher desk"