Factual error: Travis begins the movie at 26 years old, and reports leaving the army with honorable discharge in May 1973. His first diary entry just after being hired is "May 10th." In the newspapers at the end he is still 26, and it says that he has been a taxi driver for 6 months. The movie obviously does not take place in winter, and the only months referenced (plus the timeline of a presidential nomination) are June and July. Besides, 1973 would not be the right year for a story set just before a presidential election, unlike 1976 when the movie came out.
Suggested correction: This error is based on the assumption that he had just been discharged. I don't remember anything in the movie to indicate that as opposed to being discharged three years earlier.
The articles at the end of the movie say "Travis Bickle, 26, has been a taxi driver for six months since he came to New York upon leaving the Service where he fought in a special forces unit in Viet Nam" (sic). I think it's fairly obvious from the context too that he hasn't had much experience with 'real life' after 'Nam, surely not 3 years. The original script didn't have this discrepancy, by the way, because the date of his discharge was May 1971, which would account for just about enough months of difficult civilian life to get involved in the 1972 Presidential race.
Other mistake: The continuity of the sequence with Travis killing Iris' Time Keeper is choppy; audio of him screaming is heard even when his mouth is closed, and his eyes are open, then closed. He is also staying completely still for the shots of the final blow after displaying fierce blind suicidal aggression (going after a man with a gun even having no weapon and one hand turned into a pulp), where his head appears pushed against the couch in a different way between angles. (01:41:25)
Revealing mistake: Travis kills Iris' Time Keeper and fails to commit suicide. He sits by the corpse; you can steel see Murray Moston's belly move, breathing. (01:42:20)
Continuity mistake: When Travis blows the timekeeper's brains out, Jodie Foster jumps away from the couch twice in two separate shots, in two different ways. She has her arms down in the wider shot when the gun goes off, but in the close-up she is making a super derpy face with one hand in her hair before making the jump. (01:41:25)
Continuity mistake: Iris' Time Keeper zombie-crawls towards Travis; Murray Moston in the front view is holding the railing with his hand, he is not in the shots from the back. (01:40:30)
Revealing mistake: Sport initially is cautious enough to inquire Travis about a gun, and suspect him of being a policeman. Forgetting the fact that he's suddenly dumb enough to not recognize just a few days later the weirdo with a huge mole on the cheek he talked to, just because he shaved his hair (maybe he was stoned), he still is approached by a guy with a hand in his pocket, in a dark street, making weird questions, who asks him if he has a GUN, and that then says "Suck on this." That already should make him at least quite jumpy, but then, at this point Robert De Niro pulls the gun out of his pocket...but flubs it, it does not happen smoothly. For a good couple of seconds Harvey Keitel stands in front of him motionless, unfazed like no human being would, just because there is no cue for the actor to do anything but stand there like an idiot and get shot. (01:39:10)
Revealing mistake: When the Senator finishes his speech and steps off the stage, Travis begins to move making his way through the crowd. Several extras are recycled appearing in inconsistent positions - look no further than the black woman with a bright flower patterned pink shirt; she is simultaneously in front of Travis as he begins to move in the outer section, and right between Charles Palantine and the stage in the shot immediately after. And then she keeps popping up where she's not supposed to be, continuously. (01:36:20)
Visible crew/equipment: Travis Bickle got all geared up to go kill the senator. There's an opening panning shot of Charles Palantine arriving at the location. As the relatively small scope of the meeting implies, but also as shown in the rest of the scene, there's no camera crew following Palantine, no big crane rig or anything. Yet you can easily spot people in this crowd who don't care about the Senator but look at the movie camera. One of the followers is particularly hilarious; on the left of the frame, he keeps waving his hand at the camera, tries to follow Palantine and direct the gesture at him, and then goes back to signaling the camera. (01:33:50)
Revealing mistake: After taking Iris out for breakfast, Travis is in his car and looks at the building when Iris lives and/or works. And in that establishing shot, that lasts about a second fading almost immediately, from his car he is watching...himself, because it is re-used footage from the scene of one day earlier, when he was entering the building to pay for her service. The person you see go through the door, is De Niro, same shirt and jeans of that scene. Not only that, most people at the window are the same and in the exact same position, and the passersby in the street are the same. (01:18:30 - 01:29:30)
Continuity mistake: During the breakfast scene Travis calls Sport "a dope shooter." He puts his fork down saying that line (you can hear the noise too) but with the camera back on Jodie Faster, Robert De Niro's hand is at the border of the frame holding the utensil exactly like before. (01:26:20)
Continuity mistake: Iris walks into the room after Travis talked with the timekeeper. She moves past the beaded curtains and De Niro follows her. The bead strings oscillate with a difference of pace/momentum depending on the angle (they move more in the angle with camera behind Jodie Foster, because she really passes through them, and they are moving less in the shots with De Niro's face). (01:19:25)
Continuity mistake: Travis plays with his foot on the TV and ends up breaking it. The TV set falls back in a dedicated view where it choreographically crashes at an angle and explodes. However, when we see De Niro sobbing over what happened, we also can get a peek at the very right of the frame and see a very much whole TV, lying parallel to the floor, screen still lit and working. (01:14:25)
Other mistake: Travis Bickle's voiceover reading the letter to Mom and Dad says that July is Father's day month. But the holiday happens in the month of June (at least in the US). On the other hand, it's more than just a Character Error, because when you see the written part of the card held by Robert De Niro, the word used is "June" and not "July." (01:12:00 - 01:13:20)
Visible crew/equipment: Travis clears out because the policeman told him he can't park there and so he starts the car and drives. Everyone in the crowd looks (many turn around explicitly and ) as he passes by, in particular looking right where the camera car in front of the taxi is. The senator is supposed to talk in a different direction, where De Niro is looking. (01:12:55)
Continuity mistake: A policeman approaches Travis' taxi and tells him to leave; during that, as he is clapping his hands yelling "c'mon" in a super New Yorker way, a guy in a maroon shirt is pushing a cart behind the cop. Travis takes off in the next shot, and passes by, in front of him, the same guy that just went the other way. (01:12:50)
Continuity mistake: Once the store owner tells him he's gonna take care of everything and starts cursing in Spanish, Travis walks away from the store. You can see when Melio is bashing the Stick-Up Man that Travis left and nobody is in sight, but in the last overhead shot you can see someone's legs in the doorway. (01:09:40)
Continuity mistake: Taking a fatal bullet from Travis, the robber falls back. He is not just breaking the snack display cabinet, but also knocking one of the dirty Spanish magazines off the clothesline. The magazine falls with him and gets all crimpled between his arm and the counter, but vanishes entirely in the rest of the scene. (01:08:50)
Continuity mistake: Travis just fed to the security guy a phony name and address. He walks away. Look when the security guy urges the photographer to take a picture; behind the photographer there is a woman in a rather flashy orange dress, with orange headband and round sunglasses worn on top of that. But that same woman is also in front of Travis walking away, and one of the first to greet the senator. Likewise, when he shouts "Damn it!" a photographer with a brown shirt passes behind him, and it's the same one that passed behind Travis in the previous shot. (01:05:30)
Continuity mistake: In the arms dealer scene, De Niro puts the last gun down with the others he is purchasing. It's at the exact angle as the others, pointing to the right, but in the close-up of the suitcase with the holster on top of the selection, now that gun (the so called "."380 Walther") is angled left, opposite of the others. (00:57:00)
Continuity mistake: Presumably the same day when he passes by Palantine's HQ and does not see Betsy at her desk, Travis nearly bumps his taxi into the same girl (Jodie Foster) he failed to save from her pimp. The sequence does not concern itself with continuity at all; the girls move away from the taxi twice in two different ways (look at how the long haired hooker turns), and De Niro's hands change position on the wheel with no logic. (00:51:40)
