To refinish veneer furniture, lay a tarp in a well-ventilated area and place the furniture on top of the tarp. Apply wood glue beneath the peeling veneer. Press firmly the peeling pieces down to bond with the glue and dab up any excess glue. Place a heavy, smooth object on the veneer to keep it firmly pressed down during drying. Brush a varnish remover on the surface. Stain a test patch on the back of the dresser and then apply a topcoat of polyurethane. Let the polyurethane dry for at least 24 hours before handling the furniture.