Some trainers aren't set up where a mountain wheel will fit on there right. Sometimes it's a wheel size issue. Sometimes, the skewer doesn't fit into the clamp right, and the trainer may come with its own skewer, but one that's designed for a road wheel.

So, check the specs of the trainer you want to see if you get a choice without some sort of adapter or extra hassle. I also say you should put a cheap slick tire on if you go with the mountain bike.