Small pieces of loose veneer or blisters in veneer can be reglued. Large veneer repair jobs should be taken to a professional furniture repairer.
If a piece of veneer has come off the surface, lay the veneer on a flat surface and scrape off the glue. Do not wet veneer. Remove the old glue from the piece of furniture not the veneer. Put glue on both pieces and put the veneer in place, add a paper pad, and clamp down with C-clamps or lay weights on it. Be sure the pressure covers all parts of the piece of veneer and don't remove the pressure until the glue is dry.
If veneer is still attached but has loosened from an edge or corner, use a hypodermic needle or thin knife blade to insert wood glue into the area. Proceed with clamping as described above.
If you have a blister in the veneer, cut a slit with the point of a sharp thin knife at the side of the blister where the veneer is still glued. Be sure to follow the grain of the wood. Hold the slit open with the knife. Fill the blister with warm white vinegar and let it stand for several hours to dissolve the glue. Wipe away any vinegar that remains with a damp cloth. Let the blister and the surrounding wood dry thoroughly before adding glue. Then work plenty of wood glue under the blister, using a hypodermic needle or thin knife blade to get the glue in the blister. Apply pressure to flatten it. Leave it under pressure until the glue is completely dry.