In the dance sequence with the little old ladies, one sees a yellow-colored traffic signal. New York City traffic signals were dark green/gray in 1959, when the movie was set. New York would not begin using yellow until 1962, when Traffic Commissioner Henry Barnes issued an order on 17 January of that year, and this was not widespread for at least two years after that.
In the dance number with the old ladies, Bialystock and the ladies cross Fifth Avenue, which is shown as one way. Fifth Avenue did not become one way until January 14, 1966. The film takes place in 1959.
During the scene entitled "Creative Accounting" when Bialystock and Bloom are first introduced to one another, Max tells Leo to take a deep breath after nearly frightening the man into hysterics. After Leo takes his heavy, drawling breath and looks back to Max, for one split second, you can see Nathan Lane's mouth turn up in a smile, just about to laugh, and then the camera angle changes. If you watch the Outtakes in the special features, you can see the goof of Lane's in full force; unable to stop himself from laughing, which they simply cut short in the feature.
During the 'Heil Myself' number, the characters sing the line "If you're looking for a war, here's World War Two!" and make the hand gestures for 'W' 'W' '2'. The lead tenor (John Barrowman) is in a hurry to replace his hat and does the hand gestures incorrectly: 'V' 'W' '2'.
When Leo knocks on Max's door at the beginning of the movie, you see him knocking with his right hand. Later the angle changes and you see that Leo is knocking with his left hand and has a bag and coat on his right hand.
Where Max and Leo are in the park, and Max is begging Leo to do the scheme with him, the shadow of the trees changes. When the camera zooms out, the shadows are all over the ground in front of the fountain. When the camera zooms out again momentarily, the shadow is completely gone.
When Leo asks Max, how much money they put in the show, Max seems to have a heart attack, runs into the sofa and then falls to the ground. When he gets up his hair is completely messy. In the next shot when Max is opening the closet to show Leo the little old ladies his hair is suddenly completely in order.
When Max opens the safe the first time to show that it's empty, it is gray on the inside. After Ulla paints the office and everything in it white, the inside of the safe has been painted white also, even though Ulla did not have the combination.
After Max buzzes all of the little old ladies out during the "Along Came Bialy" number, the first apartment building shows a little old lady who says "Maxy" coming out second. But in the next shot this little old lady is first leading them out of the building.
The gun Franz is shooting appears to be a Luger, a very popular German pistol in the first half of the 1900s. The Luger came with an 8 round magazine and could hold an additional round in the chamber for a total of 9 shots, however 10 shots were fired without a reload. (3 in the ceiling when Franz walked in the door, 1 hit a statue, 1 hit a vase, 2 shots in the ceiling, 1 through a window, 1 hit the safe and a final shot when the gun was thrown)
The older male dancer who says "Don't be stupid, be a schmarty..." is speaking with Mel Brooks' voice on his first appearance. The next time he is heard delivering a line, he has a completely different, deep voice. However, this is deliberate: the character also lip syncs to Mel Brooks' recorded voice in the stage production.
The gun Franz is shooting appears to be a Luger, a very popular German pistol in the first half of the 1900s. The Luger was only originally designed to hold eight rounds, but can hold nine with one in the chamber. The Luger is also emptied, not jammed.
In the original 1968 film, an actor is chosen to play Hitler, and author Franz is incensed to see Hitler played for comedy on opening night, sending him into a rage. In the musical, Franz himself is cast as Hitler (until replaced at the last moment by the director) so he had to know all along how Hitler was to be portrayed. Thus, his rampage after the show makes little sense but for how "gay" Hitler ended up, but would he even know that, needing to go and have his broken leg set?