Cut off any loose sections of veneer with a utility knife. If the chip is irregularly shaped, use a metal ruler to cut it into a rectangular shape to make fitting the replacement veneer easier.
Match the veneer chip with replacement veneer, available at most hardware stores.
Outline the exposed patch of veneer onto tracing paper. Note the direction of the wood grain if applicable. Cut out the traced pattern.
Place the tracing paper cutout on top of the veneer sheet and positioned so the grain on the patch matches your notes if applicable.
Cut out the veneer with the utility knife and use the metal ruler as a guide.
Place the veneer cutout over the bare area to be patched. Trim the edges for a better fit if necessary.
Coat the back of the veneer cutout with a thin layer of wood glue or epoxy. Press the cutout onto the patch and hold it in place until the glue starts to set. If necessary and possible, tighten a clamp onto the patch to hold it in place.