Aim a heat gun at the wood veneer and slide the edge of the putty knife under the veneer. As the heat pushes under the veneer, it loosens the original glue. Move the heat gun to the edge of the veneer and aim the heat under the wood.
Lift up on the putty knife as the heat slowly softens and melts the glue. Push the putty knife along the wood veneer until it lifts up from the furniture. Scrape the knife along the top of the furniture, breaking the seal between the furniture and the veneer.
Wear goggles, rubber gloves and a respirator or mask as you apply an even layer of adhesive remover to the furniture. Peel the veneer back and apply the adhesive remover with a paintbrush. Scrape off any remaining paint with the putty knife and wipe the furniture down with a damp cloth.
Apply wood putty to any gouges or chips in the furniture. Gouges keep the veneer from attaching smoothly to the piece. Once the putty dries, rub off any excess putty with medium-grit sandpaper. Sand the surface with fine-grit sandpaper, which gives your furniture a smooth surface.
Paint a thin layer of wood glue on the surface of the furniture. Carefully press the wood veneer down and run your hands across the top, pushing out any air bubbles. Apply clamps to the top of the veneer, which keeps it pressed against the glue. Wipe a damp cloth over the edge, catching any glue that seeps out. After 24 hours, remove the clamps.