It depends on your goal. The "right" way to fix it would be to strip the old damaged veneer and re-veneer the piece.
If it isn't a valuable piece and you don't want to remove and redo the veneer, there are a couple potential options depending on the type and amount of damage.
- veneer is worn through somewhere in the middle: carve out the defect and inlay a nice design (or you could put in an ugly football patch like they do with the faces of cheap plywood)
- edges are chipped on a tabletop: trim back the edge of the table then re-veneer and/or refinish the edges
- trim back the veneer to remove the defects (and on adjacent or corresponding parts for consistency), then add a new strip of veneer or edge detail
- corners are chipped (e.g., around the sides of a leg or sides of the piece itself):
- chamfer the corners then veneer and/or finish the newly-exposed surface
- trim back the veneer to remove the defects (and on adjacent or corresponding parts for consistency), then add a new strip of veneer or edge detail
- add "corner protectors" over the corners, hiding the defects
It will be next to impossible to perfectly match the grain and insert or butt up a separate piece of veneer alongside the original, so you'll probably want to go with a complementing color or pattern if doing an inlay or trimming back and reveneering the edges.
Keep in mind if the piece is valuable, these patch jobs will only decrease its value in the eyes of anyone seeking an authentic antique. But if the piece does not have any antique value, you may be able to use these types of repairs to add some new and interesting design elements.