As mentioned, careless use of the ports blows the CIA chips... That was far and away the most common cause of trouble back when these puppies were new.
Nowadays, so many issues also depend on the model. A few common ones I see all the time:
A1200's often have Alice video corruption issues. (Heat? Pure white?)
A1200's blow power supplies.
A4000D's suffer bad caps, especially in Audio and on the 3640 board.
A4000D's can have delicate SIMM sockets.
A500's have floppy drive issues. (though, in fairness, this may just be due to how often they played autoboot games... :-P )
And, of course, as stated... LOOK OUT FOR NiCAD BATTERIES!
They show up all over Amigas, and are deadly.