(at around 1h 30 mins) When they are on the outside of the private plane to Mumbai, there is a sound of the jet engine in idle mode. In the last shot of the sequence the sound changes to jet engine throttling. However we see that the engine blades are standing still and not rotating, hence engine is not running and cannot produce that sound.
(at around 1h 45 mins) When Ethan Hunt asks Benji Dunn to "pinpoint Hendricks's location", Benji says "Hendricks is signaling from a state run TV station". However, earlier we saw Hendricks had entered the "Sun TV Network" station which is a private Television network.
(at around 57 mins) When Hunt and his team are in the Hotel at Burj Khalifa, they see the sandstorm approaching from the front of the building yet when Hunt and Wistrom exit out front (at around 1h 16 mins), the storm hits the building from the back.
(at around 59 mins) During the climbing scene in Dubai, Ethan falls many stories down the building when his laser gun sparks as he attempts to access the control room for the first time. In the next shot Ethan is already back at the control room without having to climb 20 stories back up.
(at around 18 mins) Mr. Hunt is issued with a fake identity card which helps him entering Kremlin. When he holds it in his hands for the first time the name given on it says Fedorov. When going through the check point in Kremlin and the security guard struggles to find it on a system but eventually does, it shows a different name Samokhvalov (read in Russian).
(at around 59 mins) At the end of the climbing the tower scene, Ethan Hunt securely places his hand in order to provide leverage to kick in the window. As the view rotates between outside and in, his hand moves from on the crack, to above it.
(at around 57 mins) When Ethan is hanging on the outside of the hotel, he can see the giant sand storm coming, it is reflected on the hotel windows. A couple of shots later, you still see the reflection on the windows, but now there is no storm visible.
(at around 1h 55 mins) In the climax at Mumbai valet parking scene, the wheel on BMW which Hunt uses to jump along with it to get the briefcase is on the left side. In India the wheel is always on the right side.
(at around 1h 20 mins) In the car chase scene in Dubai, Ethan's BMW Six series is shown to be a six speed manual, because there are three pedals and oftentimes he can be seen changing gear manually, however towards the end of the chase right before he deliberately crashes the car, when what would be the gear knob was shown in an overhead shot it bore the unmistakable large "D", "R", and "N" of an automatic.
(at around 52 mins) When driving into Dubai the team are in a Land Rover Defender 90. In some shots the driver's window is a slanted, sliding type found on military Land Rovers while in other shots it's is the civilian wind-down type.
(at around 12 mins) When Carter and Hunt sit in the van and Carter updates Hunt on what happened to the colleague in Budapest, some camera shots on her show a silver reading lamp coming from top right of the picture, however several other camera shots do not show this lamp, while the perspective has not changed significantly.
In the Mumbai valet parking scene, the order and position of some of the parked cars changes between shots. The high angle shot shows a Ferrari Enzo, Mercedes Benz SL 65 AMG Black, and Bugatti Veyron parked in front of the landscaping. In the moving-camera shot, a fraction of a second later, those three cars have been moved forward and a McLaren F1 is parked behind the Enzo.
(at around 14 mins) The initial scene in Budapest is supposed to take place in or near Keleti station but instead it is shot at Central Station in Prague, Czech Republic. This can be validated by the "City Elefant" trains which are in use only in Greater Prague area.
(at around 1h 35 mins) After the plane lands, they have supposedly arrived in Mumbai. When Ethan Hunt pulls up to Brij Nath's house, you can see both the architecture as well as the environment is middle-eastern (most likely Dubai), not Indian.
At 1:11 the Budapest Keleti station is visible. At 1:24 he jumps from the roof at the Vytopna Zlichov in Prague (Prague Castle visible in background). At 1:43 he is at another location around Prague, because the train passing by (type CityElephant) runs at that time only around Pague and there are only two tracks.
(at around 43 mins) Hunt and Brandt are shown at the bottom of a river as bullets whiz by from multiple villains firing at them from above. However, the program MythBusters: Bullet Proof Water, 360 Degree Swing demonstrated that high velocity rounds slow down substantially and frequently fragment with a couple feet of the surface.
(at around 39 mins) When Ethan asks William Brandt to identify the man Ethan drew on his hand while in the van, Brandt states that the subject has an IQ of 190. This is impossible. Standardized intellectual assessments have an "upper end" of only 160 points and no professional intellectual assessment will result in a finding of "IQ=190." Anything higher than an IQ of 160 could only be suggested as being, "in the 98%ile or higher."
A magnetic field powerful enough to lift a human being (who was wearing a metal vest) would also wreak havoc in everything electronic nearby, also throwing into the air the very computers the team was trying to access. The average battery pack cannot supply enough power for such a move.
(at around 43 mins) Bullets are shown moving through water at high velocity, endangering Hunt and Brandt. Bullets, however, lose velocity very quickly in the mass of water. Water tanks are often used for ballistics tests for that reason.
(at around 1h 55 mins) When Ethan gets into the car on the parking garage lift, he presses "push to start". That car has a proximity system. The key fob called a "prox" must be nearby in order for the car to start.
Ballistic missiles do not fire their rockets after they reach their highest point, hence "Ballistic". All the stages burn out before reaching the highest point of the trajectory ('apogee'). They also are targeted much higher than the ground-level burst apparently intended in the film.
(at around 1h 1 min) When Ethan is escaping the server room by running down the front of the Burj Khalifa he is using a technique called "aussie rappelling". In order to do this the belay device the rope passes through must be positioned at the back of the harness and not on the side as shown in the film. If it was attached this way you would end up going down the building side first and not face first.
(at around 1h 19 mins) During the Dubai sandstorm scene, Ethan is seen driving what appears, from the foot well angle, to be a BMW 6-series convertible equipped with a three-pedal, traditional manual transmission. No BMWs exported out of Germany have this transmission; all of them have an 8-speed automatic. Further, in a brief shot immediately before Ethan crashes the vehicle, an angle from behind the car shows that he is using the 8-speed automatic.
It is not physically possible to make a nuclear sub launch a missile within the depicted time frame of 30 seconds, or even a few minutes. That is because it is not physically possible to communicate quickly with submerged submarines. A sub on deterrence patrol is submerged about 200 meters or more where you can only reach it via ELF (extremely low frequency), allowing communication only on the order of a few characters per minute. This is used to transmit a pre-arranged code group to make the sub rise to a much shallower depth where faster means of communication can be utilised. That depth change alone takes tens of minutes.
(at around 47 mins) When Kurt Hendricks is talking about nuclear weapons in the TV-interview he is speaking in Swedish to the Russians. The Russians in the background are seen wearing translation headsets.
(at around 1h) Ethan Hunt damages first his right sticky glove (on the outside of the building, where he throws it away) and then the left (smashing through the window). In the server room, however, he has gloves on both hands again; that's because we see him remove the left glove and put on a new pair of plain leather gloves from his pack.
(at around 1h 13 mins) After the exchange of the diamonds for the launch codes, right before Jane Carter catches up with Sabine Moreau, Moreau waits impatiently for the elevator and pushes the button. The whole wall shakes under the pressure of her pushing the button, showing that it is a set.
(at around 1h 49 mins) The Nuclear Launch Device used by Hendricks has an incomplete Russian keyboard (missing four letters which are normally placed on right hand side on QWERTY keyboard buttons with left square bracket, right square bracket, typewriter apostrophe, and slash), and the User Interface is very poorly translated: the title (the inscription above the code) should say Kod podtverzhdenia, not Udostoveritelnyj kod (the word Udostoveritelnyj does not exist in Russian at all, although has some recognizable Russia parts), there is an untranslated word ("dismantled"), and the syntax is awful. (A little earlier in the movie the UI in Engish is shown, yet it of Benji's device, not Hendricks' one.)
(at around 53 mins) When Ethan is approaching Dubai by road, he is driving on the right hand side of the road. When they reach the Burj, there is a panoramic view of Dubai, the traffic is seen driving on left.
One of the characters is seen on several occasions using a MacBook Air. Although the character is actively working on the computer, the Apple logo on the cover of the machine is dark. In normal use of the MacBook Air, the logo would be lighted.
(at around 1h 50 mins) The nuclear warhead would not have re-entered the Earth's atmosphere with the final stage of the missile still attached. The nose cone would have separated in space and the warhead would then detach from it. Like astronauts in a re-entry capsule, the warhead is a 're-entry' vehicle.