Anachronisms
Mia Farrow continues to wear her [extremely well executed] 1920 fashions well into the mid-1930s, by which time women's styles had changed dramatically in the real world.
Stock footage, particularly of crowd scenes, from the mid-to-late 1920s, continues into the 1930s decade.
Factual errors
The speaking person in his 60s in one of the modern interviews in the film is subtitled as "Former SS-Obergruppenführer Oswald Pohl". If the interviews were conducted in the early 1980s, the person is evidently too young; the real Pohl was born in June 1892, so he would have been in his late 80s/early 90s at the time - of course if he had not been hanged for war crimes in 1951.
Last name of Broadway impresario Florenz Ziegfeld is misspelled (Zeigfeld) on title card for Pathe newsreel.
A presidential pardon would not clear Zelig of the state-level crimes of which he was convicted.
Revealing mistakes
At one point, the narrator mentions that there is a photo of Leonard Zelig as Pagliacci. "Pagliacci" is the Italian plural for "clowns". The correct way to say it would be " a photo of Leonard Zelig as Pagliaccio", or "a photo of Leonard Zelig as Canio (the leading male character, who plays Pagliaccio in a play-within-the-play in the opera "Pagliacci".).
