Damage the car? Nope. Damage the dongle? Possibly, but unlikely. Drain the battery? Maybe not, but o you really want to find out first thing in the morning when you're late for that vital meeting/interview, the rest of the household and everyone in 100 yards every direction have already left for work, leaving you abandoned with a flat battery and no credit on your phone because you were chatting up that hotty from the night before....
I have noticed 'some' bluetooth dongles seem to get very warm while they're in use, I would be nervous about leaving those 24/7.