Revealing mistake: When McClane goes into the icy room in the power station, look at the pipe to the side of the acrobat: it's in shot a few seconds and you can see that the writing on the pipe is reversed, revealing that in the editing room, the shot was flipped.
Continuity mistake: John's face is all bloody after he rolls from the car and "kills a helicopter with a car". The blood on his face constantly changes during the shots. Sometimes there is blood all over his face, sometimes there is quite a bit less. Sometimes he has blood streaming from the cut on his nose, sometimes there is nothing on his nose. The blood changes back and forth, so it's not just him wiping the blood away from his face (and his hands are by his sides during all the shots). (00:44:35 - 00:45:30)
Continuity mistake: At the end of the scene where Bruce Willis's own car is involved with a crash between a dustcart and taxi you can see his driver's door opening during the aerial shot after he has left the location. (00:24:25)
Continuity mistake: When McClane and Farrell are fired upon during the escort to DHS, Agent Johnson is killed and slumps against the steering wheel, and in the next shot the deceased's foot slips off the brake so the car jolts forward and hits the FBI's SUV in front of them. Four shots later, in the interior shot from the back seat directly behind McClane ducking inside the car, we see Agent Johnson sitting upright very much alive, at the left side of the screen. (00:38:35)
Continuity mistake: In the scene where McClane rebuffs Ferrell for thinking it would be cool to do the firesale, you can see pipes for firehoses right behind McClane. When the shot changes to a long alley shot, the pipes are a good ten feet away, but when it reverts back to a closeup, the pipes are back to right behind McClane. (00:55:20)
Trivia: Bonnie Bedelia makes a cameo as John's ex-wife, Holly, via a picture of her driver's license. To illustrate the divorce, Holly even uses her maiden name, Gennero. (00:36:55)
Continuity mistake: When the truck crashes into the pillar that McClane and Matt are hiding behind in the tunnel, we see a large portion of the pillar fall off from the left side. However, when McClane gets up to drive the police car, the pillar is in perfect shape. (00:41:30)
Continuity mistake: When McClane knocks the fridge over so it's on its side near the start, his head is above the top (side) of the fridge. Camera cuts and it is well below the top (side) of the fridge. (00:12:45)
Continuity mistake: During the traffic jam scene, when the truck crashes against the taxi, the view inside McClane's car shows the taxi colliding against the car, dragging it away violently. From the aerial angle, the taxi doesn't touch McClane's car at all. (00:22:35)
Continuity mistake: In the traffic jam in the beginning of the film (right after all the lights have changed to green), the car next to McClane's is black/dark. But a few seconds later when you see it from a helicopter, it has changed to white. (00:22:30 - 00:23:30)
Continuity mistake: When McClane is driving the police down the ramp to the tunnel so he can elude the helicopter, the car's windshield doesn't have any bullet holes or cracks in it, even though it did just a few seconds earlier. (00:37:55)
Deliberate mistake: While McClane drives in the tunnel, Gabriel orders, "Open the gates on the north side entrance." A few shots later the signal light changes from red to green and the north gate opens. The "warning" sign is backwards in this flipped shot. Then, when Gabriel opens the gates at the south entrance, there's a proper shot of the "south" gate opening. The two shots of the "north" and "south" gates opening are the same, but one is reversed. All the dirty smudges on the signs, booths, etc, are identical. (00:41:20 - 00:42:05)
Continuity mistake: While McClane and Farrell are in FBI Police car 279, the windshield's rearview mirror repeatedly vanishes and reappears. (00:38:25)
Chosen answer: In 2005, Hurricane Katrina ravaged the city of New Orleans, with many of the residents temporarily housed in the Superdome. FEMA (Federal Emergency Management Agency) was highly criticized for their response (or lack of), including how long it took to supply the Superdome with adequate water and food. Matt's pointing out to John how the government isn't nearly as capable of responding to disaster as people think.
OneHappyHusky