Don,
Is this a no crank situation or no start?
The TSM completes the ground for the starter relay Mike. So if the alarm is triggered (along with the siren going off [if you have the siren]) the starter does not engage. The next statement is assumption based on how I'd expect the circuit design to be laid out with that assumption being that its fail-safe mode would do the same thing in the case of failure.
That speaks only to the "security" side of the TSSM though; or TSM for bikes without the alarm function. We also know that either TSSM or TSM can shut off ignition because what the old farts among us will remember as the bank angle sensor is also built in the TSSM or TSM. And we know that function shuts off ignition because the same modules were used on both fuel injected bikes and bikes with carburetors.
Then, for anyone thinking "I'll just eliminate it" in case of some future failure there are also the turn signals and hazard lights to consider. Even if you did wire around all that you'd have an annoying check engine light because the now absent module wouldn't be sending it's state-of-health signal at ignition on. On the newer bikes I believe the neutral and clutch interlocks are also managed through the -07 and later module.
So in a pinch could you hard-wire a ground to the start relay (that's #9 and #12 on the module if I remember correctly). I'm about 95% sure that would let the bike start and run (pre -07 modules anyway; the -07 module for the later bikes is still available new though). But you'd lose bank angle sensor safety, turn signals, and hazards. Not ideal but might at least get you home or until such a time as a replacement module was scored on eBay. All of that potential annoyance is why I like spare parts if I'm keeping my old bikes. Even a spare TSM would let you continue a road trip only losing alarm function. Granted, the pairing process for TSM or TSSM to ECM is a pain-in-the-ass doing it at the handlebars. It's much easier at a dealership with their tools though. So even if you fail in a parking lot or on the side of the road you're only delayed long enough to get towed in.