Before I say anything - yes what I'm about to say is a bit dumb and this is not an endorsement.
Out of pure curiosity, I've run my Volt "out" of gas several times. I say "out" because it will actually turn the engine off while there is still a tiny bit of fuel left in the tank. (Confirmed by popping the gas tank, taking the cap off, putting it back on and closing it without putting any gas in the tank - the Volt wil fire up the ICE the next time you start it and will run for a minute or so before shutting off again citing a lack of gasoline). One of these times I decided just to leave the tank empty and drive the car. I did so for several weeks and several hundred miles. While it will ding like mad at you when you turn the car on, it will operate (granted in reduced power mode - which at max output is about normal driving style for me, so no impediment) for the full ~40 miles of range. (Got down to 1 mile on the guess-o-meter once with an empty gas tank - first and only time I've had range anxiety in a Volt! ;-)). So, to whatever extent this scenario is akin to your what-if situation, I would say the Volt operates just fine with no operable generate/engine. Granted, nothing was technically broken in my example (no check engine light) so I can't vouch for the same scenario if the MIL was illuminated.