The answer is a qualified "no"; in most of his many origin stories, the Joker looks the way he does because he was permanently disfigured, and his skin and hair were permanently discolored.
As we can see in the following pictures, which are taken from Alan Moore's brilliant masterpiece The Killing Joke, Batman identifies an impostor when the man's makeup rubs off on Batman's glove:
A few quotes from around the interwebs:
The Joker began as a very dark character. Originally, it was unclear if the Joker’s pale skin and green hair was done with make-up or was simply how his skin looked. Over time, it seemed clear that it was his natural state.
Eleven years after he was first introduced, the Joker was given an official origin story. How he got the natural clown look was revealed to be the byproduct of his swimming through a vat of chemicals when he began his criminal career as the Red Hood. The story where this was revealed was actually a great one as it featured Batman looking into just who the Red Hood was and why he disappeared. Actual clues were littered throughout the story and the Joker’s reveal was a real surprise.
Nowhere in this, or any other, story is the Joker given a real name. He is always simply “The Joker” who once went by the name "Red Hood".
The first depiction of the Joker's origin story; Moore revisited this idea in his The Killing Joke
"No recounting of the Joker's origin has been definitive, however, as he has been portrayed as lying so often about his former life that he himself is confused as to what actually happened.
As he says in The Killing Joke:
"Sometimes I remember it one way, sometimes another... if I'm going to have a past, I prefer it to be multiple choice!""
"They've given many origins of the Joker, how he came to be. That doesn't seem to matter—just how he is now. I never intended to give a reason for his appearance. We discussed that and Bill [Finger] and I never wanted to change it at that time. I thought—and he agreed—that it takes away some of the essential mystery."
– Jerry Robinson, the Joker's creator