How to Remove Wood Veneer from Furniture... VERY informative instructions!!! First step, wet the wood. I just poured water over the top of the dresser and smoothed it into the wood. The gloves come in handy, let it soak in. Then I wet the towel, not so it is soaking, but pretty damp is good, lay the towel over the veneer you want to remove, working in small sections, and you simply lay the iron on the towel for a bit.