Judging from your picture, I would remove the rest of the veneer. Older pieces sometimes have nice wood that can be refinished. If it is nothing special, but sturdy, paint is an option. You could also apply new veneer. The reason I would remove the rest of the veneer is it usually continues to lift, and you won't have an even surface if it is left on.
I agree remove old veneer and sand. I would paint. Use oil based primer , lightly sand again and paint with oil based paint.
that is such a large irregular surface that you would have to square it off in order to replace missing veneer. But veneer is funny, it can be removed or loosen easily in some areas and a bear to get off in other areas. I guess you can best determine which way would be less work, remove all or replace some.
Remove the damaged veneer. Sand down the under wood if it is in good condition and stain or paint. If the under wood is not in great condition. Buy some MDF or Grade A plywood, get it cut to fit over the top of the table. They will do that at Lowes and Home Depot. Attach to top and paint. Use a sprayer to avoid brush marks. Also remember to top coat.
I have to agree with these ladies. This veneer is shot. I'd remove it, sand, and paint. The table looks as if it's still in usable condition so it would be a shame to just toss it.
You can glue on some 1/4" thick birch, maple, or oak plywood if you plan to stain. You can use thin underlayment if you plan to paint. You can try to remove the veneer. Either heat it up with a blow dryer to soften the glue and peel it off. Or wet it down with hot water and peel it off.