Sand or scrape the back of the piece of veneer from the hole to remove old glue and embedded dirt.
Stain the veneer piece to match the furniture, if necessary.
Apply a thin coat of wood glue to the back of the piece. Push the piece of veneer in place and allow the glue to dry for 24 hours.
Sand the area carefully with the fine sandpaper. Apply additional stain if required.
Cut a veneer piece from a section of the furniture that is not visible with a razor blade, like the inside of a leg, cutting the piece slightly larger than the hole.
Carefully trim the veneer piece to the size and shape of the hole.
Sand the back of the piece and apply a coat of wood glue. Push the piece in place and allow the glue to dry for 24 hours.
Sand the area carefully with the fine sandpaper. Apply additional stain if required.