During the bike chase when Ethan and Benji are in the car, but before Ethan asks if his seat belt is on, Benji's seat belt alternates between being on and off.
After escaping from the water-cooled storage tank, Ilsa is wearing a tank top and bra with both sets of straps visible at the shoulder. When she turns to change out of her wet clothes, she removes the tank top and has no bra on.
When the opera Turandot starts, there seems to be a continuous action sequence of chasing and fighting in the stage loft area that lasts about 5 minutes, but over an hour and a half of the opera takes place during that same time up to the point when Nessum Dorma is sung, not to mention that it is sung after the intermission.
During the motorcycle chase, as the bikes are heading uphill, the headlights on Ilsa's bike change sides. In most shots the round light is on the rider's right side but in one shot the film has obviously been flopped and the round headlight is on Ilsa's left.
When Benji is kidnapped in Paddington Station, Ethan tells Luther to watch Ilsa as he and William run to the garage. As Luther makes haste towards Ilsa, in the first shot, there is a woman sitting alone in the background. In the subsequent shots of the woman, there is a man who shows up looking towards Luther.
The wetsuit that shows blood oxygen saturation, worn by Hunt when infiltrating the submerged data center in Morocco, is showing unrealistic measurements. When breathing regularly, the blood oxygen level is always at around 100%, so two or three deep breaths are not going to trigger a significant rise, as shown in the movie. Also, even after a 5-minute breath hold the blood oxygen level never drops below 70%.
When entering the building with the cooling system, Ethan breaks a glass window with a vibration or sound device. In the close-up before, metal wires for reinforcement are visible in the glass, which also creates the checkerboard pattern. This wire is not destructible in that way.
While using a pay phone, Ethan orders the other end to "go secure." To do that, both ends must be encrypted. It's impossible to go secure if one end is using a public phone with no encryption function.
In the opening scene, where the artillery shells are loaded onto the aircraft, a biological warfare symbol is seen on the outside of each shell. When Ethan Hunt receives his debrief it is stated that the shells contained VX nerve agent which is a chemical warfare agent.
Benji refers to Austria's Chancellor as Head of State, when in fact he is Head of Government. The President is Head of State.
Metal detectors would only prevent infiltrators from using oxygen tanks made of metal. All they'd have to do would be manufacture some that were made of plastic. They'd only need to last a few minutes, so they would not have to contain a lot of overpressure, hence they would not need to be very strong.
They are disavowed agents racing against time to get what they need, they don't really have the resources to be "manufacturing" non-metal oxygen tanks.
Ethan mentioned that Secretary of the World Bank was on a plane that disappeared. While there is no Secretary of the World Bank, but rather a President, the IMF movies obviously takes place in an alternate world to the movie viewers, where technology and acts such as happen in the movies are real. In this alternate world, it is quite possible that the World Bank does have a Secretary.
When Ethan is climbing over the plane's wing at the beginning, he jumps onto it from some dirt hill close to the taxiway. No airport would have a structure/hill of dirt that close to any runway. However, in the shot that follows, the plane is shown to be taxiing down a runway, immediately followed by a shot showing it back on the same taxiway as before.
At the power plant in Morocco, Benji greets the gate guard with the Arabic Muslim greeting "As-salamu alaykum". The region around Casablanca is mostly Berber, some Tuareg, not Arab at all. Regardless, Arabic is an official language of Morocco, and it would be fitting for an obvious foreigner to use Arabic or French instead of Berber.
Even the most powerful computers on earth do not need elaborate water cooling as shown in this film. All these computers need is an air conditioning system.
It's not only a cooling system but also a security feature hence the included metal detection feature.
The opening is captioned as taking place in Minsk, Belarus where Ethan Hunt chases the military transport plane and is hanging on to the door during takeoff. As the plane gains altitude you can see a road in the distance and traffic is seen to be driving on the left, indicating this was shot in the UK. In Belarus traffic drives on the right much like the rest of continental Europe.
When Ethan answers a call from +0 (Lane) it's in a correct manner using a Lumia 930, however when he's talking to Lane, he's holding the phone the wrong way (upside down). The speaker on the back of the Lumia is visible while he's holding it, and on that particular phone the speaker is on the bottom part of the back of the phone, so the phone is upside-down.
The street scenes after the Opera shooting are meant to be set in Vienna, but it's quite clear from the street signs and markings that these scenes are actually shot in the UK, and at one point the emergency vehicles can be seen driving past a Be At One cocktail bar, which is a UK only chain. A few shots later, Ethan, Ilsa and Benji also drive past a Bank of China branch which is located next to the Bank of England in the City of London.
At the beginning, during the plane scene, the plane is Russian, the instructions that Benji is reading are in Russian and yet, when the plane pilot handles the cockpit instruments all the writings are in English.
As Benji is trying on the mask in front of the mirror, which is actually not a mirror but just a glass window, the man playing his 'image' is not synchronized with him. His 'image' has his hands at a lower level while touching his cheeks.
During the motorcycle chase supposedly set in Casablanca some of the filming locations are actually in Rabat (notably the exterior wall of 'Kasbah Oudaias'). Immediately following this, during the dramatic reverse crash, the car lands in the square outside the Hassan II Mosque in Casablanca, almost 100km away.
In Vienna, Benji gets off the U-Bahn at Schottenring station. While getting instructions from Ethan, he is seen taking two escalators up and finally surfacing at the Karlsplatz station, which is a different station by the Opera building.
Ilsa Faust crosses Westminster Bridge towards the South Bank, yet in the next shot she is walking from Lambeth Bridge towards Westminster Bridge to meet Atlee.
When Ethan explained about the airplane crash involving the World Bank secretary to Benji, he said that the plane disappeared in the Pacific Ocean, while the article displayed on the computer showed a picture of the Indian Ocean.
When Ethan and Benji discuss the escape of Syndicate throughout the globe, Jakarta is indicated on the map with a longitude and latitude of 6 17 45 S, 106 02 27 E. These coordinates actually belong to the Province of Banten, near the town of Anyer and the tourist spot of Karang Bolong, which are both too far from Jakarta. The correct longitude and latitude of Jakarta are 6 12 0 S, 106 49 0 E.
All the water flooding in for the cooling system, which takes Ethan into the chamber, must also flow out again somewhere at the same flow rate. When the flow restarted, he should have been swept out of the chamber while unconscious.
Lane knows what is actually in the secure computer facility. But Ilsa believes it to be a stolen ledger (probably false information planted by Lane) and informs Ethan of this. They have a motive to steal it and use it against Lane, but as far as they know, Lane simply doesn't want anyone else to have it. Ilsa knocks out Benji and steals the USB stick to return it to Lane to continue to gain favor, following her MI6 orders. But Benji knows he has a copy which could be used to bring down Lane, so there's no reason for Benji and Ethan to chase after Ilsa to prevent Lane from recovering what they believe to be his own ledgers.
Before Ethan jumps into the cooling system, he's told that the system can detect metal. Hence, he cannot bring in oxygen tanks. We see Ethan inside the system wearing a sleeve with a computer-based screen showing his vital signs. This screen and computer would not be possible without metal.
Ethan starts his stopwatch a full 20 seconds before he enters the water in the cooling system. Not a very accurate way to measure how much time he has left.
Several characters refer to the data storage device as a disk. It is clearly a USB drive AKA a flash drive.