I have a trainer where the tire rests on the drum - this is the first time I've had this type of trainer in my 20 years of cycling. I'm not sure what kind of noise you are getting, but I got on my trainer after not riding it for a couple of weeks this morning and my tire was making all kinds of noise. All of a sudden it came to me that it was probably because I had a low tire.

And, so, I got the pump and pumped the tire up to its recommended psi and what do you know, the noise stopped!