Welcome to the board so sorry you had to come with a problem that might be hard to try and fix. First off, does it stay tacky all the time or just when hot?
Do you have any leftover from that shirt or do you wish to try risking your iron to get it to happen on another piece?
On my samples, I'd try my standards -- nail polish remover for try one, and peanut butter for try two... then testing the fabric first, WD40 for try three, is amazing what that stuff can fix. If I couldn't get a good fix with those the last fix would be a piece of bridal illusion netting over the logo. You can either cover the whole square or cut it out around the logo and zig zag it down and make it look deliberate. Will probably still attract pet hair and have some sticky to it but is possible.
Use a clean hand towel for rubbing in whatever you try. Paper towel will shred and stick to the sticky. Check the towel frequently (as well as where you are rubbing) to see what (if anything) is coming off and the effect on the logo.
Even though the quilt is together you can place another piece of fabric over the problem piece, either a different one or maybe put a label or memorial there. Blind stitch it into the seams and then quilt over. If you had to, you could even hand quilt from the back over the existing quilting pattern, but many of us have had to take out substantial amounts of quilting to redo before and in that case, might as well put in a new block if you have one.
Most likely after whatever treatment you will have to give it a washing before giving it. That alone might solve the problem or it might make it worse and the sticky will attract lint.
If this is any consolation at all, I once made a quilt for a friend with an unstable maroon fabric that turned the sophisticated blue, silver, grey and maroon quilt with sparkles of white with non-sparkles of baby pink. And to make matters worse, some poly-blend whites made it in and they did stay bright and sparkling. My friend, because she was a true friend, said -- oh look, a quilt I can feel good about drinking grape juice in bed.
PS: Accidents happen and it can be hard on us when it is our fault. But you are doing a kind thing and memories are still there in the quilt. Some of us have loved ones that are a bit difficult and might even appreciate them still being difficult in the quilt.
Best of luck and don't feel too bad.
Last edited by Iceblossom; 10-09-2019 at 06:14 AM.