I'm assuming that you're referring to foundation piecing, and not English piecing over papers? If it's foundation piecing, I've seen some lovely and intricate quilts where instead of using papers the foundation is made from fabric, either something like plain calico or a non-fusible interfacing. The foundation isn't removed and becomes part of the quilt.
I will probably be proved wrong by someone, but I imagine if you use paper it would be very difficult to remove without distorting the hand stitching.