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.
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: 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)
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)
Other mistake: The man (turns out to be a mafia bigshot) who goes to Iris' pimp Sport asks him if she's in the usual building, and walks off, no doubt about his intentions. It's broad daylight, and it's summer. Yet Travis, who is also on booze and pills after failing to kill the senator and surely did not bid his time, arrives in East 13th Street only when it's already pitch black outside, and the john is still with Iris in the 'work' room, with them being fully clothed and with the bed made (but she's on her knees so they are 'just about to do it'). (01:37:30 - 01:39:45)
Other mistake: Travis built a special arm holster for a gun that allows him to pull a gun easily, popping it in hand while his arm is already outstretched. It's the weapon he prepared for in the iconic "You talkin' to me?" scene, and practices handshaking with it. We see a scene when he prepares to go kill the Senator cutting the sleeve of the shirt off. And yet, when the actual attempted hit is shown, Travis has his hand (the gun hand) inside his jacket, alerting security. (01:36:30)