What would most likely burn out on a dryer that was hooked up to a clogged exhaust vent and ran for several months??

Actually, I'm not sure that anything is "most likely". The house might burn down.

There are thermal switches that will turn off the gas when the system gets to a maximum temperature. That would keep the dryer from damaging itself. I don't think that these would be exceptionally likely to fail, however. If they did, the system would overheat.

The biggest problem is that the clothes won't get dry.