Make sure the filler can be painted or stained.
If the veneer has chips missing, fill the area with a wood filler such as DAP Wood Filler if you plan on applying a paint or stain finish. If there's nothing to clamp to, place wood over the veneer and weigh it down with paint cans or something heavy.' The scrap piece of wood keeps the clamp from damaging the veneer and applies even pressure. Place a wood scrap or shim over the veneer and clamp it down. 'Use your syringe to squirt some wood glue under the veneer, then press the veneer down. 'When it's lifting, first clean out any debris between the veneer and the surface underneath,' Dvorak says.
Over time, the veneer can come loose from the underlying wood surface or chip off. Even dressers, tables, and desks made from real wood probably also have veneers.