Continuity mistake: In the scene at the diner, where Travis puts a soluble tablet in a glass of water, you can see a plate of open cheeseburger in front of him. But Travis never ordered, nor received any cheeseburgers. He only ordered and received a cup of coffee only. (00:17:10)
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.
Continuity mistake: While De Niro is talking to Jodie Foster you can see a bus out the window. When the screen switches to Jodie Foster's view you can see the same bus driving, but then it changes back to De Niro's view you see the same bus driving up to the window again. This keeps happening throughout the scene, the bus keeps driving around the building. (01:27:00)
Continuity mistake: When Iris and Travis are having breakfast together, at the beginning of the scene, she is wearing green sunglasses. She eventually takes them off, but when she puts the sunglasses back on, the sunglasses are now purple with a gold rim. (01:25:00 - 01:29:00)
Continuity mistake: When Robert De Niro shoots the pimp's hand off in the hallway (shortly before shooting Sport the second time), blood splatters on his face, across his left eye and forehead. As he stumbles up the stairs (still shooting the handless pimp), the blood has changed, and finally in the shot where De Niro pantomimes shooting himself, the only blood on his face is on his chin. (01:39:40 - 01:42:40)
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)
Continuity mistake: In the scene where De Niro shoots and kills the man robbing the convenience store, the clerk takes De Niro's gun off of him and starts beating the already dead body with a pole. If you watch the scene carefully, the dead body changes positions contrary to how the man is hitting him. Clearly the scene was shot several times with the dead body being placed in different positions. (01:09:40)
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.
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: 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)
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)
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)
