We got an older foose ball table and I noticed that the fake wood paper veneer is dried and peeling. I want to take this paper off and varnish it. I have tried using an iron and steam, and I tried scoring it and using wallpaper remover but that doesn't seem to work. Any other suggestions would be most helpful.

If you have a fake wood paper veneer, then you likely have particle board (otherwise known as MDF) under it. And THAT will be butt ugly varnished. You would be better off refinishing it in a similar fashion... like a simulated wood-grain Formica type product. Not hard to work with as I'll bet its all flat surfaces.

Not particle board, the legs are solid wood, and the body is furniture plywood

Hair dryer aimed at it an inch away as you scrape, no steam.