Again, thank you all for your wonderful suggestions! So far I have tried rubbing alcohol which just removed the top layer of the residue. Then I tried nail polish remover which removed another layer or two. I washed the area with mild soap after trying both of these. I can still feel a "sticky" area when I put my finger on the residue area, so will have to keep trying some of the other suggestions. This stuff is really embedded in the fibers of the cotton. What a learning curve -- will never use masking tape again. I now have a set of "pin on" numbers and letters to use when keeping track of block arrangements (you can get them at Joann's). I'm off to the store to find another product to try, and I'll keep you posted.
Thank you for the suggestion of snowballing the corners, but that just would not work in this situation as the background with the residue morphs in the side setting pieces that become a part of the quilt. If I ever get this quilt finished, I will post a picture.