A friend has a generator where one phase has stopped working. The phase not working is the one supplying the 240v output. The 3-phase does not work. If the 240v outlet is switched to one of the working phases to supply 240v, will running the generator like this cause further damage? 3-phase not required
It depends on how it's failed. If it's shorted internally, then it'll heat up and catch fire - which wouldn't be ideal. If it's failed open circuit, then it shouldn't do any further damage, it's just a dead leg. You'll lose some efficiency (i.e. excessive fuel consumption), as the engine is dragging a dead phase around, but assuming this is just an occasional use set, that's probably not an issue.
All three phases should be giving out the same voltage - i.e. 220 across the board, or 240. The specs for 240v AC power actually allow anything from 220 to 250, so even at 220 you're (just!) in spec.
Machine Mart occasionally sell reasonably priced 3-phase alternators (I think they're Mecc Alte branded), they also do 240v ones, but for some reason they're more expensive, IIRC.