Continuity mistake: When Satine says, "It's lady's choice.", from a front view, the Argentinian is very close to the bearded man, but from a back shot they are farther away. (00:18:45)
Continuity mistake: During the "Like a Virgin" sequence, when the Duke is sitting on a couch, his hair changes between the shots: From loose to perfectly brushed and straight.
Continuity mistake: During Satine's first number, Diamonds, when she is on the trapeze, her lipstick is a bright red. It changes to a dark pink and then back to red during the song and is finally pink when she has passed out.
Continuity mistake: During the "Rhythm of the Night" dance sequence you see The Petite Princess singing on the bar in a hindu outfit behind the gnarly men. After that you see her in her can can skirt. Then any time you see the gnarly men she's back in the hindu outfit.
Continuity mistake: In the end of "Your Song" when Satine and Christian return to the Elephant Room, Satine puts her right arm around Christian's neck. Shots from Satine's right side reveal her right forearm (you can see her elbow), but shots from her left side show that her sleeve cover the entire forearm, and uncover it when she doesn't put her arm in Christian's neck anymore (touch his shoulder).
Continuity mistake: When Christian and Satine are up on the elephant and she is about to leave, you see him put his hand on the pole. Then it cuts to another shot and you see him put his hand on it again in a different way.
Answer: There is actually a deleted scene from the movie that explains this part. Zidler has run the Moulin Rouge into heavy debt because of his obsession with electric lighting, and the club will fold if he can't come up with the money to cover that debt. That is why he's so desperate to have Satine come through and secure the Duke's patronage. They're all sunk without that investment.