Joker's paint was just war paint, similar to how Jack attempts to disguise himself to the leaders of the mob in "Batman". He is also shown without it at the bank robbery when he reveals himself to the bank manager. Theatrics, consider the paint be the Joker's "mask".
nope, it was not. The Joker's face was bleached by the chemicals. In the scene at the restaurant/museum, the Joker seems to appear without make-up but when Vicky Vale throws the spaghettis at the Joker's face (when he screams "I'm melting! I'm melting!") you can see his real skin color, white that is.