I have a 4 poster bed and a vanity with the veneer peeling off. I would like to remove it and paint the furniture.
Sand lightly what is peeling off and then get some zipit stripper and leave on until you see it bubbling and scrape off with a putty knife.
I have a great post that I wrote: https://karupp-did.net/fetor-to-fetching/
In that post, I tell all the issues I had with removing veneer and I found a way that was sooooo much easier. Here is the Hometalk post that is a little condensed version of the above post from my blog. Good luck!
A heat gun or hot hair dryer can help in the removal process.
Here is a post I wrote about removing the veneer from an old dresser https://www.homeroad.net/2012/06/determination-and-dresser.html