If the veneer is fine maybe it would be possible to re-glue and clamp? Find a local trim carpenter, furniture repair person or cabinet builder/shop before painting. Seen a few pairs of Dunlavy’s with the lifted veneer, so apparently Mr. Dunlavy’s cabinet maker used poor quality glue or just the wrong type.
The problem of painting, unless you spend the money to have a automotive body shop do the work is, it probably will not look nice or hold up well. If you do decide to pull the veneer off, you’ll likely have to sand the rounded side edges as those are hardwood and flush with the veneer edge, so keep that in mind, then you’ll need to sand off any glue or mastic left and hopefully the veneer doesn’t partially tear on you, leaving areas where it’s glued very well to the MDF.
The cabinet box is simple MDF which isn’t going to be as durable as the oak veneer for taking abuse. I think you may find in the end to remove the veneer, sand, prime and paint will be a lot of work and if you’re not a pro the outcome may be less than desirable. I’d strongly suggest to find a local cabinet maker or etc. and hire someone to do it right, and likely it will be better than new and retain their value. Good luck.
