I remember in Gotham Central where he got shot several times through the chest and was still alive and conscious albeit paralyzed on the ground. There is definitely some healing factor going on. I would say it would take something like a high octane lethal injection or a 100000 volt chair to properly kill him and be done.
Recently, I have been giving much thought to a 'what if Joker dies?' scenario and the results were actually interesting. Too bad the writers never dares to go into that direction.