I'm working on a Judy Niemeyer paper pieced quilt right now. According to her directions, if you don't get all the tiny pieces out it's okay, because they will dissolve when the quilt is washed and become part of the quilt. I find that I can get 99.9% of the paper out by scratching the intersection with a nail (and I do NOT have long nails). I do use a short stitch when paper piecing.
Oops, just read your post again and think you may be using the tissue paper on the top of the quilt when you're quilting it. I think the same is true, and the paper will dissolve and go away. You could also spray it with water and see if it's easier to remove then.