I don't know about "unjustly". Firing an employee for posting a customer's signature on the Internet seems absolutely fair.
If the signature had been cropped out, leaving just the note, and the waitress was still dismissed then all this outrage would have been understandable.
I've been on a project like this-- they ganged up on me, got me kicked off the team and then turned on each other.
I went back 6 months later and the last 2 remaining guys had overengineered something that the customer hated but they were happy with, and they'd taken so long to do it there was no money left for any sort of frontend.
I imagine you could filter out the 48-52Hz range (probably less) and make your fake recording unverifiable.
Taking the audio from the time you want to claim the recording took place and adding it shouldn't be too difficult either, though that might leave other clues that it's not genuine.
If the signature had been cropped out, leaving just the note, and the waitress was still dismissed then all this outrage would have been understandable.