I've never tried, honestly. Probably because I always wanted it to have one, usually the Wings card.
Anyway, the way those things are designed, the Personality Card slot is essentially a PCI card (with a separate PCI channel for the modem). Since it's designed to be interchangeable with at least three separate cards, there's nothing on any one card that's required to be present for the Mac to run.
I can't guarantee that the machine will boot and work like it should, but it won't explode or anything; at most it just won't boot. If it does boot, it may be a little screwy: no Macs are supposed to be missing the audio section, so software might get a little confuzzled, especially under OS 9.
Don't think about trying this on an AIO, though, because those use the personality card for video passthru and display control.