There is a product called Citra-Solve that dissolves gummy,gluey, sticky stuff. i have used it on fabric and it didn't hurt or fade it.

Try Drafting tape. I use it to mount photos. It's like masking tape, but no residue.
https://en.wikipedia.org/wiki/Drafting_tape

That was my thought although it would change the design of the quilt. Good luck with the outcome.

I have both Goo Be Gone, and Undu. I would try the UnDu first. I don't know if you remember seeing an add with a sticker on a piece of toilet paper. They sprayed it on the toilet paper and the sticker came off without breaking the toilet paper! It says Use on Paper, glass, plastic, walls, most clothing, Finished leather, vinyl, upjholstery, carpet, wood furniture, and more--then it says it might harm unfinished leather, or suede. I think I got it at Michaels in the scrapbooking area.

I guess you know now that masking tape is a not the stuff to use on fabric-------or much of anything else. Next time buy some painter's tape. It doesn't stick real tight, but it's great to use on fabric.

I read somewhere that WD40 was good for removing tape residue from fabric. However I would certainly try it on a separte piece to see if it was going to cause a stain or not.. They claimed it wouldn't, but I would have to try it first.

Lighter fluid takes residue of everything much better than goo-gone. My Father used this since "forever". It takes residue off fabric (he used it on his silk ties). I use it on fabric, glass, cardboard, slick paper etc. Of course, like everything you haven't used before I would try it on some sabric first "just to be sure". Good luck.

I never use masking tape. I use painters tape.Lighter fuild should work.

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.

No quick answer to that but suggest next time you elmer glue little number slips on or pin