While John and Jack held up at the Ball room by Alik with their hands tied, Jack cuts his way using the knife he had hidden in his socks. We see John jumping at Alik with his hands tied and in the next scene, his hands are untied and has got hold of a gun.
When John and Jack are stealing a car outside a nightclub, they are trying to pick the lock of a Maserati Quattroporte. Even the car key John produces, and the subsequent shot of a trunk full of guns is that of a Quattroporte. However in the following shots, they are driving a Maybach 57, because they open several different cars and the Maybach emblem can be seen on the trunk lid before they close it and drive away.
After the breakout from the courthouse Jack and Komarov are handcuffed when they are in the blue van. When John stops the van and they get out, the handcuffs are gone.
(at around 20 mins) When McClane is chasing after Jack, at the moment when McClane rams the truck chasing Jack and says, "Jack! I'm not done talking to you!" there is a brief shot of Jack driving the blue van with the driver's side on the right side of the van. In all other scenes, the driver's side of the van is on the left.
When John is saying goodbye to his daughter his hair is a rough buzz cut of a few millimetres, at the start of the chase scene in Russia it's still fuzzy, but when he stops during the chase, in a building, his head is close shaved and shiny. John's hair cut changes throughout the movie.
The villains have a chemical spray that "cancels radioactivity". Completely impossible.
Vehicles throughout the film have license plates with letters unique to the Russian language, but no such letters are used on license plates.
The boxes used to store and move the enriched uranium have both the radiation symbol (correct) and the biohazard symbol (incorrect) on them. Biohazard symbols are used to indicate materials of biological origin.
The "Reaper" drone mentioned in the first action sequence refers to the General Atomics MQ-9 Reaper. It is a propeller-powered drone, unlike the one shown in the film. The drone in the film looks more like the jet-powered Northrop Grumman MQ-4C Triton.
(at around 30 mins) After Jack picks up John on the bridge, the scene switches to Chagarin on the phone about to be informed that Komarov has escaped. The conversation begins with Chagarin saying, "The elections are next week," but the translation is rendered as "The appointment is next week."
During the car chase John is called twice by Lucy and the same clip of him saying, "I'll have to call you back," is used both times. It's obviously out of place the second time because he shouldn't be going downhill.
After bribing the hotel staffer for the access card John McClane swipes the card in the elevator with the reader strip faced out so that the machine would be unable to read the card.
Komarov, who is played by a German actor, speaks Russian with a thick German accent.
In one of the aerial shots of the Cougar HEV chasing the blue van down the narrow side streets, the cars parked on either side of the street move before the HEV actually hits them.
The travel guide Lucy gives to John ("idiot's Travel Guide") says it comes "with new routs", not "with new routes". It also says it was written by Tom Karnowski, who served as Executive Producer on this film.
Early in the film when John (Bruce Willis) is walking to the court house after leaving the taxi. You can see multiple protesters waving French flags as well as Russian flags.
When the small blue car (Renault Clio) is hit during the chase in the beginning of the movie, the quad or motorcycle which is carrying the cameraman is visible for a short moment.
After a fight, both main characters Travel to Chernobyl, fully armed, by freeway having a light conversation. Chernobyl is not in Russia but is actually a territory of Ukraine. It is more than one evening's drive, there are no freeways and two fully armed Americans would not be able to make this crossing from one country to another with out being stopped.
During the chase scene on the highway Hungarian ads for IKEA and Árkád can be seen on light poles.
When McClane arrives at the airport Hungarian and European Union flags are visible in the background.
When Jack and Komarov are brought to the court house, someone by the fence is waving a flag. The colors are correct Russian colors, but the flag is hanged wrong way to the pole.
All roads leading into the Chernobyl/Pripyat exclusion zone have military checkpoints - yet neither the villains or the heroes seem to have any trouble gaining access.
CIA (or otherwise) operated US drones would not be allowed, or even able, to perform surveillance flights over Moscow, Russia's seat of government and home to both the Russian Ministry of Defense and the Russian Air Force's headquarters.
During the shoot-out in the warehouse in Chernobyl, a hundred or more rounds are fired and not one of the blue fluorescent tubes marking the entrance is broken.
Jack is on a carefully-timed CIA mission, with a drone standing by. But he only escapes from the courthouse when the gang attacks, something the CIA couldn't plan for. The drone is a pivotal part of Jack's mission, but there is no explanation how it will help him escape Moscow.
Grenoble is mentioned by McClane as being a Swiss city. It is actually in France, more than 100km on the other side of the Swiss/French border.
During the chase from prison, John stops the van facing the armored truck. John moves out of the way at the last moment so the truck hits the other cars with full speed. If the van was still there, probably John and Komarov would die due to the impact of the crash. So the truck should aim to stop right in front of them to block their escape rather than ramming them if they wanted them alive.
John McClane repeatedly exclaims that he's on vacation, which is a completely false statement. He's gone to Russia to help his son, not for a holiday.
As a supposed law enforcement officer, John McClane breaks the law a lot more than he enforces it (by stealing vehicles, damaging property, assaulting innocent civilians, endangering the public, carrying guns across an international border, etc). Not that he'd have any jurisdiction anyway, as an American cop on vacation in Russia.