When Nikki and Ali are sitting at their tables in the dressing room, Ali looks into the mirror and sees Nikki brushing her hair. When the camera turns to Nikki, who yells at a girl for touching her things, the camera then turns back on Ali and Nikki is still brushing her hair even though she had stopped.
You can hear feedback when Ali is onstage and Nikki sabotages the speakers. There wouldn't be feedback because the speakers have been disconnected. Feedback is usually caused by microphones, not speaker cables. Additionally, since the dancers were supposed to be lip-syncing at this point, the microphones on stage would not be live. Even a voice as strong as Ali's would not be heard over the noise in the bar or the band playing.
During the "Tough Lover" sequence, a crew member lowers the curtain by pulling on the side of the rope closest to him. When Tess tells him to raise the curtain, he begins to pull up on the same portion of the rope. This would be impossible, due to the way fly systems are weighted. The proper method is to pull down on the other side of the rope.
The goof item below may give away important plot points.
The premise of the plot rests partly on Ali replacing a newly-pregnant dancer, Georgia. The movie's time line takes course over two months (or less), and in that short time, Georgia becomes pregnant, engaged and married (in that order), also returning to perform with Ali at the Burlesque Lounge during the finale (supposedly back from maternity leave, which would be impossible). This contradicts the plot line in which Tess must come up with enough money to save the club in thirty days. (In the same day, she sells the air rights above the Burlesque Lounge and buys out her partner's share, Jack finishes writing his song and Ali performs it, during which a whippet-thin Georgia appears as a dancer despite showing during the wedding a few days earlier.)