Continuity mistake: During the fight in the end, Travis shoots a man's hand and his four fingers are blown apart; yet when the same man jumps at Travis as he enters Jodie Foster's room, you can briefly see two of his "missing" fingers on Travis' shoulder. (01:39:40)
Visible crew/equipment: Right before De Niro's character is shot in the neck, you can see the small charge used to detonate the blood. (01:39:50)
Revealing mistake: When De Niro gets shot in the neck, his bald skin cap wrinkles considerably when he presses the wound with his hand. Confirmed by make-up artist Dick Smith in the making-of documentary. (01:40:00)
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)
Continuity mistake: When Travis gets hit point-blank in the arm, there's a close-up of the gun falling down the stairs, which is re-established at the end when the camera pans back through the corridor showing the mayhem. But if you look at the shot that immediately follows that close-up, you can see Robert De Niro dropping the gun, that just bounces next to him, by the wall. (01:40:40)
Continuity mistake: The John stumbles inside the room in his dying moments and falls back with his arms spread out. In the shot from an elevated angle, with Iris screaming, you can see he is a certain distance from the radiator (and Iris' feet). In the POV shot from the door he is much closer, and when Travis is tackled by the mutilated timekeeper the dead man, Peter Savage has both arms up. (01:40:50)
Continuity mistake: In the close-up of the stabbing, Murray Moston's hand looks different compared to the previous shot. (01:41:20)
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)
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)
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)
Character mistake: The "Taxi Driver Hero to Recover" newspaper clipping misspells "Pittsburgh" at the end of the first paragraph as "Pittsburg." (01:46:55)
Continuity mistake: During the final cab ride, Robert De Niro in close-ups has a distinctive curl of hair across the forehead. In the close-ups of his reflection in the mirror it is not so, and same for the dialogue with Betsy. (01:48:10)
Continuity mistake: In the scene where the man has DeNiro pull over to the curb and says he is going to kill his wife with a .44 gun, there is a close up of the meter clicking over to $2.75. In the next shot from the backseat, the meter reads $0.65.
Suggested correction: The reason for that is because after Travis stopped the car, he turned off the meter. Then Scorcese's character asks what he's doing, and to put the meter back on. The $.65 indicates "$.65 first 1/6 mile" as clearly painted on the cab.
As the text of the original mistake stated, there is a close-up of the meter. A biiig one, that follows by quite a few seconds the meter being turned off. You can see the 0.65 before that close-up, you can see it after, it then changes to 0.75 and so on. This correction is totally wrong and the original post is correct.
Character mistake: When Travis is buying guns, Easy Andy (the gun salesman) makes a few errors. Firstly, he states that the Smith and Wesson 61 Escort is a "Colt .25 Automatic." Secondly, he calls the Astra Constable a ".380 Walther." And lastly, he talks about the ".380 Walther" saying "during World War II, they used this gun to replace the P38. Just given out to officers." This is also incorrect.
Other mistake: There seems to be no bullet impact into the wall or near Murray Moston after the .44 Magnum revolver is discharged. Given the general direction and angle the weapon was pointed, a bullet should have blown out part of the wall near Murray Moston.
Other mistake: Travis' hair length appears inconsistent throughout the movie; he begins working in early May with his hair of a certain length, his hair appears trimmed differently in certain scenes driving the cab, it is certainly much shorter, sort of a buzz cut, during the scenes when he interacts with the senator's guards, but it is already longer when he is doing the "You talkin' to me scene" and the murder of Stick-Up Man, which is in between the buzzcut scenes. And then of course, there's the ending.
Continuity mistake: During the same scene there is a white van parked across the street behind De Niro. Then it's gone. Then it's back.
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.
Continuity mistake: Travis' hair is medium length for most of the movie. But in some scenes (his chat with the Secret Service agent and Iris following) De Niro's hair looks shorter. Probably these scenes were filmed after the mohawk scenes, without enough time for his hair to regrow fully.
Continuity mistake: The guns Travis purchases from Easy Andy are (regardless of what Andy calls them), from left to right, a Smith and Wesson Model 29, a Smith and Wesson Model 36, a Smith and Wesson Escort, an Astra Constable. In the overhead shots at the gun range when Travis goes to practice immediately after, some of the guns change: the first 44 Magnum is the same, but the 38 becomes a Colt Detective Special, the S&W Escort a Galesi-Brescia Brevetto 5, the Astra a Walther PPK (which is what Andy called it earlier).
