I use good quality 200-250 thread count sheets. (Not sateen unless for backing) for background fabric- whites, for example on a 30's quilt. I almost always use sheets for backing. I go to a discount store like Ross or TJM or Marshalls, and see what they have.
I machine quilt mine on a frame. Used to just machine quilt on a short arm machine. Never had an issue. Our grandmas used all kinds of stuff in their quilts. If there is a big difference in the fabrics, like a homespun and a 250 thread count, you may have some issues, but I have no issues using sheets with quality quilting fabrics at all.
The quilt police sometimes have a different view- but i say go for it.