Might try WD-40. I've had good luck with it.
Thanks. Whatever I do, I'm going to test on a piece of the same material. My buddy who does corporate interior design, said do nothing. He says the material the hook is stuck on is a thin sheet of paper laminated with a photograph of wood and over it and a very thin layer ( if any ) of a protectant over that. He warns that any chemical can cause damage.
What I didn't tell you earlier is that I already tried to remove the hook ( it's supposed to just pull off, using the tab ). The tab broke off, so then I used a non-serrated butter knife to try to pry it up. Got that started and lifted the rest, and a chunk of "wood grain". My attempt to glue it back in place with white glue only resulted in it tearing and bunching up ( thinner than a sheet of toilet paper ).
So, my solution was going to be to add a larger hook to cover the damage and remove the hooks on either side, which is how this started in the first place ( I had too many hooks, too close together ).
It appears that the simplest thing to do is put the hook back in place.
My advice is don't get carried away with those convenient self-adhesive hooks.