Okay, so I think I'm ready to revisit the antique dresser. This was one of the most difficult pieces I have ever refinished. It was very large and very beat up. I had never worked with a piece that had peeling veneer, so I was unsure of what to do with it. Should I replace it, pry it off or patch it with some wood putty? Turns out I did two of the above. Some parts were too far gone to try and repair, so they required complete removal, other places I fixed with wood filler. Prying off old veneer sounds like it'd be relatively easy, but I was wrong. It took lots of elbow grease, a hammer and a blade to get that stuff off. I had to chisel away slowly in order to get it off in one piece, otherwise it would split and that was not good.
So I ended up prying off several pieces on the drawers and on the inner side.