You could try gluing it and clamping but it looks like getting a clamp on it might be hard and I have not had good luck gluing veneer that is peeling away like that. That is cheap and easy to try so I would start there though. Some of the other suggestions to repair it are good as well. I will throw out another, buy a sheet of veneer and cut out a square an inch or so larger than the area that needs to be repaired. Lay the square over the area to be repaired and cut out the bad area in the shape of the repair piece, then glue the repair piece in, a little wood putty and sanding to match might do the trick.
Depending on where it is peeling away, you might try putting on small decorative trim pieces stained or painted in a complementing color. I did this all along the underside of our countertop as decoration and to make some new cabinets blend in with older existing ones.