Continuity mistake: Indians are on a hilltop and a jet vapour trail can be seen in the sky.
Deliberate mistake: Blofeld doesn't recognize James Bond in this film, even though they met face-to-face in the previous movie, "You Only Live Twice." There is a production-related reason for this. Ian Fleming wrote "On Her Majesty's Secret Service" in 1963 (in which Bond and Blofeld met for the first time), and he wrote "You Only Live Twice" in 1964. However, "You Only Live Twice" was adapted for film first (in 1967), and "On Her Majesty's Secret Service" was adapted afterward (in 1969). Because the 1969 film was so faithful to its source material, Blofeld and Bond are basically meeting for the first time... again. The producers were aware of this continuity problem and intended to have James Bond undergo plastic surgery for "On Her Majesty's Secret Service" (which would conveniently explain Blofeld not recognizing him, as well as the fact that Sean Connery had been replaced by George Lazenby in the lead role). But the plastic surgery idea was discarded in faithfulness to the novel, resulting in a glaring continuity problem between the 1967 and 1969 films.
Visible crew/equipment: When Woody Allen cuts out a piece of glass out of a shop window, the camera, dolly, some crew members and a crowd of spectators can be seen reflected in the glass.
Factual error: In the 19th Street Parade, it shows the Budweiser Clydesdales pulling their famous wagon. Even though this movie is set in 1897, Budweiser did not start using Clydesdales until April 1933 when the sons gave their father, August Busch Sr., the owner of Budweiser, a 6 team team of Clydesdales to celebrate the end of Prohibition.
Other mistake: The split second that a running student passes in front of her, Miss Mackay calls her name and restrains her, reacting much faster than humanly possible. (00:02:50)
Continuity mistake: At the end of the town collapse, a sincerely drunk Ben dives head-on into a tub where a lady (in her underclothes?) is just enjoying a bath. Propelled by Ben's momentum the tub with both passengers slides down the slippery planks and splashes into the river, while Ben rearranges himself into a comfortable sitting position between shots. If I wasn't completely mistaken also the lady changes her outfit, since the embroidery is missing when they float along. (02:23:35)
Visible crew/equipment: When Neil and Brenda are in the row boat, the bulge of the mic transmitter is visible at Brenda's lower back. (00:43:45)