The Veneer on my chair peeling off. How do I restore it?

Peel off what you can. Use a hot iron over the veneer to soften the glue and it should peel off easily.

Depending on how bad the veneer is, you can remove it or stick it back down. Check with someone at your home store for advice on the type of adhesive and tools you'll need. Bring pictures.

If you want to remove it completely, what's is the base made out of? There are solvents that will help lift it off but you probably don't want to damage the rest of the chair. Is it just a bit or a lot? We need a little more info for accurate advice.

If you want to save the veneer you should be able to glue it back down. Apply wood glue, hold down the veneer with clamps or something heavy until it drys. Hopefully it stays down. If it doesn't then you will have to replace the veneer. They sell veneer strips at most big box home improvement stores.

Since this is a porch chair, I have a suggestion. After doing one of the above suggestions, or both suggestion, here is mine. Buy a roll of 3-ply natural jute, and an all purpose wood glue. Put glue on the end of one leg at a time and wrap the leg with the twine, being careful not to overlap as you travel up the leg as far as you want to go. Cut, then tape the end to the chair until totally Dry. Then you can remove tape. Do other three legs I. The same way. A very decorative way to solve the problem, and inexpensive.

HEAT WILL LOOSEN THE GLUE.