Is There An Apostrophe In The Phrase New Teachers Luncheon?

Yes, there should normally be an apostrophe in this phrase. As I suppose that a lot of teachers are invited to this luncheon, the apostrophe should go after the "s" in teachers: New Teachers' Luncheon. (If it was just one teacher, then I suppose you would say "New Teacher's Luncheon," although it sounds a bit odd.)

There is one way you could leave out the apostrophe, and that would be if you made the phrase into a kind of instruction, ie "New Teachers: Luncheon." (The same way you might put, for instance, "New Students: Register at this Desk.) However, it sounds a bit stern and formal for what is presumably meant to be a nice invitation!

Basically, if you want to say that this luncheon is FOR new teachers, you need the apostrophe. It's really the same as saying "Children's clothes" or "Young Farmers'  Club." Hope this helps.