It's not so much the full size spare, on some AWD/4WD vehicles even a 50% worn tread is enough of a difference to cause damage to the differentials or transfer case.
The owner's manual for the car will spell out what you can and cannot do. Some will recommend only putting the spare on the front, others will recommend disabling 4WD or the AWD system and possibly the traction/stability control as those systems can be screwed up by one wheel that's turning slightly faster than the others.