Some questions and recommendations
How old is the piano and what timber is the veneer?
I'm sorry but I have to disagree with ian above:sad1:. At least for antiques, the least modification made, least removed, is the best choice to retain value. There are many very valuable veneered antiques that have small chips repaired with small pieces of similar coloured veneers. It is almost impossible to get a perfect match but people expect and accept some signs of age. So, even if the piano is relatively modern, I recommend finding out what the timber is then finding from a commercial supplier or cutting your own piece of veneer with a bandsaw to patch/match. I do agree with ian that you should not remove a piece from elsewhere on the piano, any additional damage is additional loss of value. There are suppliers of veneers listed in woodwork magazines - for parquetry/marquetry etc. Carbatec used to sell some Australian veneers too.
Antique pianos were often veneered in mahogany or walnut.
In antique pianos the adhesive would be an animal glue of some kind, probably rabbit skin glue. You can test that with some fairly hot water on a rag. If you wipe where the glue was with the wet rag (protect adjacent good veneer with masking tape or similar). If it is animal glue it will soften and may even wipe off. If it is animal skin glue then you need to glue the replacement patch on with animal skin glue. Carbatec sell a Titebond brand liquid hide glue that I have used for such repairs. It is easy to use and has not shown any signs of failure for me. Alternatively, you can buy rabbit skin glue on eBay, soak it, heat it and use the original adhesive.
If it is a modern adhesive then you will need to scrape it off pretty thoroughly. I would still use the hide glue for the repair because it is reversible. If it isn't good enough you just apply hot water and the patch can be removed .
Good luck