Without intentionally damaging the microphone, access to the signal path coming out of it, or noisy blocking... your solutions are going to be pretty limited.
I think the best easy bet would be a noise canceling circuit (like one ear of a noise canceling headphone) and then playing white noise on the microphone side. If you've got the active canceling and decent insulation, the noise won't have to be that loud to make sure you don't have usable sound on a recording.
The much simpler method would be disabling in software, like uninstalling drivers and such, but this may be more difficult for just the mic element of a audio codec or under a lot of more closed operating systems.
If you are fine with a permanent/damaging solution and can't cut the wire/trace, then maybe a few drops of very thin super glue through the port would do the trick. It shouldn't take a lot on the diaphragm itself to stop it from moving, though you may need to try an acoustic stimulus or something to get the glue to seep in far enough.