I'm using a $20 household fan 24/7 to ventilate a damp crawl space by
sealing it against one of the crawl space vent (i.e. becomes an exhaust
fan).

Eventually this fan is going to fail, perhaps the bearings wear out and then
the fan stop spinning.

Should I worry about a fire, or do all UL listed fan have some sort of
overheat protection?

Is there some sort of fan that is designed to run 24/7 and is well protected
against overheating?