In the flashback scene of their father's suicide, the Colt .45 Commander used had its hammer closed after he shot himself. The hammer would have been in the fully cocked position after firing and can only be closed manually.
The jewelry salesman who recognizes Szell yells, "I know who you are, you murderer! I know who you are!" before Szell kills him with his retractable knife. Though we can still hear him speaking when the scene cuts to Szell slicing his throat, his lips aren't moving.
After Dr. Szell slashes the Holocaust survivor's throat with the knife that extended from the mechanism on his arm, he flags a taxi, and gets in and then wipes off the blade and retracts it back into the mechanism on his arm. But while he is flagging and entering the taxi, the knife is not visible.
The man helping the fuel oil truck driver back up into the street exhales condensation, indicating that the weather is cold. This is despite the fact that a radio announcer has previously stated that the weather is unseasonably warm.
When Rosenbaum is driving his malfunctioning car in New York (steering wheel left side) he reaches to adjust the air conditioning controls with his right hand. However, the close-up shot clearly shows a left hand adjusting the controls. (It should be noted that climate controls were commonly located on the left side of the dashboard in American cars of the '60s and '70s. Rosenbaum is driving a 1970 Chevrolet Impala or Caprice, which indeed has the heater/air conditioner controls to the left of the steering wheel.)
During one shooting scene, government agent Janeway (William Devane) is firing his revolver with his left hand. However, throughout the rest of the movie, including other shooting scenes, he uses his right hand.
When Doc dies on the floor of Babe's apartment, he has what seems to be a piece of tape stuck to the bottom of his shoe. Then, when he is next shown again with chalk outlines around him, the piece of tape is gone.
Babe holds Szell at gunpoint in the waterworks - the hammer is back (cocked) on his 1911. He fires a warning shot, the hammer is still cocked. When Szell challenges him to shoot, Babe takes the safety off and cocks the hammer back - neither action is necessary when the hammer is back.
When Babe (Hoffman) shoots Janeway (Devane) in the old house, Devane falls to the floor only slightly disturbing the rocking chair beside him. After a quick shot of Hoffman, the shot returns to the "dead" Devane and chair is rocking very visibly.
When Rosenblum pulls up behind Szell, the amount by which the cars are offset changes. First the offset is about 1/4 car width, then by 1/2 width, then by no offset when Rosenblum starts to ram into Szell.
During the master shot of the first dental torture scene, you can clearly see the "marks" - a t-shape of tape on the floor - where one of the thugs is to stand in the bottom right hand corner of the screen.
After Doc flees the Opera, when we him in the Grecian temple, dolly tracks are clearly visible on the floor. As the camera dollies in, we lose sight of them but Doc obviously high-steps over the tracks.
Dr. Szell is supposed to be living in Uruguay before going to New York. We see shots of what is supposed to be Uruguay which show people traveling in canoes through the jungle, but in Uruguay there is no jungle.
Szell's brother and Rosenbaum crash their cars into a fuel oil tanker truck, which explodes and then covers the street with a flaming oil spill. Although flammable, especially when sprayed or vaporized, home heating oil is not explosive in the manner depicted.
When the father's body is shown after his suicide with the pistol, the hammer on the pistol is down. When one shoots a pistol, the hammer remains in the cocked position until the last round when the slide remains retracted.
The Colt .45 Commander that was used was supposedly a souvenir from World War II. It's distinguished by the shorter barrel and slide and the hammer is in the shape of a ring and not a hook as is on the military M1911A1. The Commander wasn't introduced until 1949 and it was almost strictly produced for the civilian market.
During the movie, Babe (Dustin Hoffman) fires several rounds from his Colt Commander semi-automatic pistol. A real Commander would eject a fired metal cartridge case after each shot; however, no cases can be seen to eject from Babe's pistol, nor can any be heard hitting the ground.
When Elsa and Babe are at the zoo, she says the French word for "penguin" is "pengwan" (approximating French pronunciation; it is actually closer to "manchot", "penguin" being more along the lines of a puffin.
At the beginning of the film, the radio announcer notes "unusually" high temperatures, and the owner of the Impala is angry that his a/c does not work. However, he is wearing a sweater, which would suggest a cooler day. Also, the man who is guiding the fuel oil truck into the street before the crash is wearing heavy jacket, which would also suggest a much cooler day. It is later reported that this was Yom Kippur, which occurs at the beginning of fall, when a hot day is still very possible in New York City.
When Doc "breaks" into Babe's apartment, Babe grabs a flashlight and shines it around the room. It is clearly on a fixed mount for the camera as the beam is way too steady as it's shown being moved around the room.
The goof item below may give away important plot points.
When Szell's goons are trying to pry open Babe's bathroom door, at one point the entire wall and door jamb move back and forth several inches and clearly are not fixed to the floor. A moment later, as they rush through the doorway, it appears solid once again.
In one scene a man attacks Babe but Janeway saves him by stabbing that man in the back by a switchblade, but it turns out that the dead man is safe because the knife is fake. However, to make such a trick work, the knife has to slide straight and not to be folded as the knife in the film.