When Agatha, Rooster, and Lily St. Regis are singing "Easy Street", in the road, you will see Agatha beat her fuzzy rope on the road, and you will notice a very large piece of the rope falls into the street, and in the next camera change, the little fuzz piece isn't there anymore and was probably picked up by a crew member.
Before the Christmas party, Mr. Warbucks and Annie are singing and dancing; at one point during this song ("I Don't Need Anything But You") Annie's lips move as she sings her line, but no sound comes out.
During the "Easy Street" number, while dancing in the office, Miss Hannigan wears teal shoes that match her cardigan. When they move on to dancing in the street, she wears brown shoes that match her skirt.
The location of McGillicutty's Tavern moves. As the truck drives away with Annie in the laundry, it shows the tavern as being down the road to the right of the orphanage, whereas at the end of "Easy Street" when Rooster, Miss Hannigan, and Lily dance down the road and into the tavern, it is to the left of the orphanage.
When Rooster and Lily enter as Annie's parents, Lily sits on the front of Hannigan's desk. When the shot switches to Hannigan's face you can see the edge of the desk, but Lily is gone. When the shot switches back to behind the desk, Lily is there again.