When Mr. Warbucks dates the check written out to Mr. and Mrs. Ralph Mudge, it is dated for July 1933; however, Camille, the movie that Annie, Warbucks, and Grace go to see earlier in the movie, was not released until 1936.
During the musical number "Sign", Miss Hannigan's bathroom contains several large bottles of her bathtub gin, and she even drinks from one. The bottles are 16-oz. Sheaffer Skrip "master" ink bottles. However, they got their distinctive shape (similar to the CN Tower) in the 1950s. In the 1930s, Skrip master ink bottles were cylindrical, with grooves running around the top a bottom quarter.
During "You're Never Fully Dressed Without A Smile", one orphan uses a liquor bottle as a microphone, then carelessly throws it over her shoulder. It obviously falls to the floor, but doesn't shatter. Polyethylene, which is used to make most plastic bottles, was invented in 1933, and couldn't be produced in mass quantities until 1936.
Edward Herrmann's FDR states that Theodore Roosevelt is his uncle. In fact, while related to TR, his wife Eleanor is TR's niece, her father, Elliot, being TR's younger brother. FDR is, more accurately, TR's uncle-in-law (if that relationship exists)
The character of Orphan Annie is supposed to be 11 years old. However, when Daddy Warbucks takes Annie to the movies, the movie they see is "Camille", released in 1936. Later, when he is presented with Annie's forged birth certificate, it is revealed that her birthday is October 18, 1922. That would make Annie 14 years old, not 11.
During "Let's Go to the Movies" after Annie tap dances behind the curtain, Grace comes out and they continue dancing. As the camera circles around, a crew member is reflected in the vanity mirror behind them.
During "Little Girls", as Miss Hannigan shoos the orphans out of the room and the camera zooms in on her, the shadow of the boom mic coming closer is visible. In widescreen, a crew member's arm is visible, for a split second, on the right side of the screen.
When Annie approaches the balcony during "I Think I'm Gonna Like It Here", she is reflected in the safety glass across the balcony, and its edge is visible on the left. The glass vanishes in the following shot, when she receives the flowers.
A gyrocopter cannot hover or even slow down below 25 miles per hour. Like a normal airplane, it must maintain forward motion to keep the rotor spinning and keep it flying. It would be impossible to slow it enough for someone could grab a rope ladder.
In the scene where Annie accidentally breaks Oliver Warbucks' knee, Punjab places him on the couch, and the sole of Warbucks' shoe is worn out at the toe. If he's a billionaire, why are his shoes worn out? He should have thrown them away and gotten a new pair.
During "Little Girls," Miss Hannigan can be seen drinking alcohol and drunk. Since the movie is set in the summer of 1933, the 21st Amendment has not been ratified until December 5, 1933 thereby repealing the prohibition of alcohol, however, it was very common for people to make their own alcohol - "moonshine" - most often in a bathtub during the prohibition years. (During the later "Sign" number between Warbucks and Hannigan, Warbucks makes a reference to "bathtub gin" and Hannigan ends the song after being backed into a full bathtub. She splashes Warbucks in the face with the liquid from the tub. Warbucks wipes his hand down his cheek, licks his fingers, and makes a face, indicating that the bathtub is in fact full of homemade alcohol.)
Miss Hannigan falls for Rooster and Lily's ruse (until they reveal it) but then goes on to reveal that Annie's parents died in a fire. In reality, she rightfully expresses disbelief that these people are claiming to be Annie's dead parents. What she fell for was Rooster and Lily's disguises.
In the run-up to the number "It's a Hard Knock Life" the orphans bother Miss Hannigan by singing in their beds. When Miss Hannigan comes into the dormitory she says if they're singing they must not be sleepy, so she puts them to work in the middle of the night. Within minutes of the them beginning to work we can see it sunny through the window. This is due to editing: they have been working all of those hours and the film jumps several hours ahead.
At the end of the film, Annie is not rescued in a helicopter/gyro-copter. Oliver Warbucks clearly refers to it as an auto-copter throughout the film and calls it a vehicle of his own design. Though the movie is set around 1933 and modern helicopters were not designed and flown until 1936, this is a fictional film and the fictional Oliver Warbucks invented a fictional device.
Before Annie and Rooster reach the top of the B&O Bridge, Rooster grabs at one of Annie's legs, but misses. Annie stops climbing for a second, with both of her feet on the top step at the top of the screen. In the wide shot she is climbing again, with her feet on different steps.