What’s the best way to repair damaged areas of veneer on an old piano I’m restoring?

Filler rarely absorbs stain the same way as wood, so you’re wise to ask. I’d start by making some intentional dings in samples of veneered wood, then try different combinations of filler and stains that you’ll find in hardware stores. Every finishing situation is unique, and it’s important to test your recipe before tackling a high-stakes project like yours. Complete a full finishing treatment over a stained test repair before passing judgment. Staining alone over the filler won’t give the same effect as staining followed by three or four coats of urethane.

One issue to consider is the need for strong filler where the veneer is chipped on edges. Regular wood fillers aren’t all that strong and they may break in time. In my work restoring furniture, I’ve had good results using fine-grained autobody compound for filling veneered surfaces that also need to be strong. I colour the filler before applying it, so the end result is as close to the final stain colour as possible. Non-reinforced autobody filler sands well and doesn’t shrink.