What kind of paint can I use to paint over veneer?

Okay, so the process is the same, clean, repair with wood putty or glue (depending on if the veneer is loose or chipped), sand lightly making sure that repaired parts are smooth, prime your piece, let it dry then paint. I would use a satin or eggshell finish using a regular water based paint then finish it off with MinWax Polycrylic to protect it. If you want a distressed look, you could use chalk paint. With chalk paint you don't have to sand it, although I would sand the repaired parts and you don't have to prime it, you'll just use enough coats to make sure your color comes through, then lightly sand off where you want the distressed look. You'll need to wax the chalk paint when you're done and clean and wax it every couple of months.

Read the link I sent, that should help a lot. Here's one on chalk paint.

If you want some techniques with chalk paint, you should look up more articles on chalk paint techniques using your search engine. Usually Bing or Chrome.

Good luck.

Where can I find chalk paint I've looked at Walmart