A keyboard! Well, if you launched it out of a cannon I suppose it open quite a few doors...
I think that was the original answer, but 'key'board occurred to me first
A piano has many keys, but it cannot open a single door because it is a musical instrument and not a lock or a key.
The answer to the riddle is: A piano. A piano has many keys (musical keys), but each key cannot open a door as a physical key would.