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.
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.
(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.
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.
The Chernobyl reactor at Pripyat was an RBMK, a graphite-moderated reactor, which uses natural uranium, not enriched uranium. There would be no bomb-grade uranium in the reactor or produced by the reactor, only U-238 which is useless in weapons.
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.
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.
(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."
When Komarova takes Collins' cell phone to call his his daughter, Irina in the safehouse, the dial pad is clearly showing on the screen. It's clearly a touch screen smartphone running Android OS. When a call is connected on a smart phone running Android OS, the dial pad disappears and the in-call screen shows.
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 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.
The goof item below may give away important plot points.
In the end when John and Jack jump from the second floor while the helicopter crashes into the building they drop through a glass roof then into a pool of water. The camera follows them into the water. The next clip is at "pool"-level viewing the surface and the exploding helicopter outside. The surface of the pool is undisturbed.