Here's an article I found for you. I have done this, and it does work.
This link provides you some answers......hope it helps you.
First, you'll need to either glue down the loose veneer or remove it. If you remove it, you will need to fill any imperfections and sand it smooth. Then you can prime and paint.
See my post on an easy way to remove veneer:
If you want to see my full blog post I have more details on my site:
Hi Sca: Here is a site that shows how to do that. If you are going to sand the areas that are peeling, be sure to feather the edges so that you don't have bumps or ridges where the peeling area is.
Hi this is Peggy. Sorry the veneer is peeling from your furniture. I hope this will help you fix it. Stay well. How to Refurbish Particle Board Furniture With Paint
www.thespruce.com › Home Decor › DIY Projects › Refurbishing
Jun 27, 2019 - Painting particle board furniture is an inexpensive upgrade. ... Over time, or with especially hard use, the plastic veneers can get scratched, ... Peeling veneer: The glues holding plastic veneer to the particleboard core can ...