When the Narrator is quitting his job, the thermostat alternates between white and black as the shot cuts back and forth between him and his boss. (The 2009 Blu-ray release digitally alters the scene so the thermostat is now consistently white.)
When Marla is on the phone with the narrator and exclaims that she has attempted suicide by a Xanax overdose, the camera zooms in on the prescription bottle clearly labeled "Xanax 300mg". Xanax is only sold as .25mg,.5mg, 1mg and 2mg tablets. The 300mg dose does not exist. Also, the pill spilling out of the bottle is .5mg pill.
When Tyler is discussing the reason oxygen masks are on airplanes he is incorrect. Breathing 100% oxygen does not create a state of euphoria. In a loss of cabin pressurization, it is hypoxia (lack of oxygen) that can induce euphoria. However, Tyler never states as a fact that oxygen produces euphoria, it is simply his interpretation of the safety notices, and his explanation for why the people in the pictures have such calm expressions.
After the scene in the bathroom where Tyler threatens the police chief, as the waiters are emerging into the parking lot, you see Bob's (Meat Loaf) pants fall down, briefly revealing the body suit he was wearing.
The goof items below may give away important plot points.
When the Narrator is breaking into the building (near the end of the movie), he slams a bench into the glass door, and it rebounds hits him. He eventually shoots the door he tried ramming, and when he kicks the glass and goes through, the bench is gone.
Most "continuity errors" are explained or implied within the film itself as 1) clues to the twist ending, 2) manifestations of the Narrator's confused, disoriented view of the world, 3) the sense that prankster Tyler Durden has tampered with this film, or 4) all of the above.
When you see that Bob is dead on the table, as the narrator is talking, some viewers have reported that you can see his body breathing. However, the apparent breathing results from the swinging of the light fixture and besides, the body used for the shot was actually a life-size doll.
At the end of the movie, while the narrator is sitting in the chair and Tyler is holding a gun, Tyler lights up and takes a cigarette out of a pack, flips it with his fingers and puts it in his mouth. In a real situation, no cigarette packs are packed tobacco side up, so some say the pack Tyler was using had to be pre-packed for the shot. However, some smokers flip a single cigarette upside down and leave it in the pack until it is the last one left. This is known as a "wish smoke". It is possible that Tyler/writers/director did this on purpose given the situation.
When Tyler splices porno frames into family films, the audio and picture tracks would have only a barely noticeable 1/24 of a second interruption (as is shown). However, the image flash and sound flash are shown as happening simultaneously, when in fact, if they were spliced into the same frame on the reel (which is what we see Tyler doing), they would be 21 frames apart (i.e. the single image frame would occur about a second prior to the single audio frame). The sound for a 35mm film is read by a head located about a foot past the projection lamp; so, for the film and sound to remain in perfect sync, the soundtrack for a given frame is located about a foot in front of that frame, that way the image and the sound are projected simultaneously. For the sound and image flashes to occur at the same precise moment, Tyler would thus have to splice the frame of audio onto the reel about a foot further along than he splices the image frame, which he doesn't do (not to mention, it would be impossible to perfectly sync it by hand anyway). However anything we see Tyler doing must be taken with a pinch of salt, since he is just a part of the Narrators imagination. It's probable that since he imagined his alter ego to be someone who is everything he wanted to be (handsome, good in bed etc) he also imagined him to be able to perform impossible feats for his entertainment.
In the scene when Project Mayhem is returning from destroying the Coffee Shop, Bob is shown as being shot in the head. When the Space Monkeys uncover his body, there is a massive exit wound at the back of his head. In the flashback sequence when he is shot however, the police officer is clearly behind him when shooting, and Bob falls forward chest-first, as if getting shot in the back. The exit wound from a hollow-point bullet, which most police officers carry, would have a small entrance wound (in this case, the back of Bob's head) and a large exit wound (the front of Bob's head). This is not the case when looking at Bob's uncovered body.