In the first film: When Alma Dray is reading the newspaper in Paris, she reads an old article about Lionel Shrike's death happening in the 70s. But in the beginning of the sequel Lionel Shrike's death takes place in 1984.
Dylan has a cut lip and bruised eye/cheek when he is pulled from the water. In the next shot the bruises are completely gone. A few moments later at the magic shop he's sitting with a blanket wrapped around him and all of the bruises have returned.
The clock in Thaddeus Bradley's cell : 7h52 at the beginning of the scene, then 7h58, then 7h52 again at the end of the scene.
Thaddeus tells Dylan to take him out of the prison just before 8 p.m., but a video recording of them leaving the prison has a time signature of 14:10, which is 2:10 p.m., nearly six hours before Dylan even arrived.
At 37:02, Mabry goes to sit between the three horsemen sitting on a lounge. Atlas is sitting on a lounge to the right. As the shot changes to show Mabry forcing himself between the horsemen, Atlas has magically joined the other horsemen on the same lounge.
On the UK mainland, the FBI would - at best - only have a consultancy or advisory capacity. Any law-enforcement would be in the hands of the UK Police. In London, this would be the either the Metropolitan Police, or the City Of London Police. There would certainly NOT be a blue US-type van parked on a London street with FBI plastered all over it.
The movie shows the FBI operating in London in order to capture the Horsemen. The FBI does not have jurisdiction to operate outside the United States, although they can act with agencies in other countries.
Twice in a movie, the character falls into water inside a metal safe box. On both occasions this fall is significant - ~60 ft (~20 m) in the beginning, and ~100 ft (~30 m) mid-movie. The safe box has a broad bottom and stops almost immediately, as seen in the movie. The magician inside would crash at the bottom of the box with nearly the same deceleration as if he fell on a hard surface. Such a fall would kill or badly injure him.
When Walter is speaking to the horsemen, he speaks of a high-tech piece of hardware, however, instead of referring to it's ability as "decrypt", he refers to it incorrectly as "unencrypt". For someone who's supposedly ahead of the rest of the world in technology, not sounding very tech-savvy.
When it's 00:00 on January 1st, it's not night in Sydney, it's 11 AM. When they 'reveal' stuff at the end of the movie, and they show that things happen exactly at 00:00 in the night in London, it can't possibly be dark in Sydney, Australia.
In Now You See Me (2013) it is quite clearly stated that The Eye only recruits new members every 50 years, and yet Lula May (Lizzy Caplan) was recruited just 18 months after the rest of The Horsemen. However, it was quite clearly stated that Henley decided to leave the Horsemen, and since there must be four, she was replaced by Lula.
In Now You See Me (2013), the audience is shown a flashback of a young Dylan saying goodbye to his father Lionel Shrike (Elias Koteas) as he leaves to perform the trick that will kill him. It is daytime, Lionel is shown as a late middle aged man with receding hair and he leaves Dylan at home. In this sequel, this whole event is completely ret-conned so that the trick takes place at night, Dylan is present at the location where the trick is being performed and Lionel Shrike is shown as a much younger man (Richard Laing) with a full head of hair.
CORRECTION: The scene that this refers to is only in the theatrical version/an alternate ending of the first film. The actual scene only has Dylan explaining how everything was connected and him giving a brief explanation of what happened to his dad and the safe. Since the scene is only an alternate/extended ending, the opening scene to this film is canon where as the extended ending of the first film may not be canon.
CORRECTION: The scene that this refers to is only in the theatrical version/an alternate ending of the first film. The actual scene only has Dylan explaining how everything was connected and him giving a brief explanation of what happened to his dad and the safe. Since the scene is only an alternate/extended ending, the opening scene to this film is canon where as the extended ending of the first film may not be canon.
In the magic shop in Macau, both Morgan Freeman and Mark Ruffalo's characters refer to speaking Mandarin. In Macau, they speak Cantonese Chinese, not Mandarin Chinese.
The "Stopping the rain" trick that Daniel Atlas does is impossible with rain, as it doesn't fall down in linear streams. The line of drops that seems to float in midair earlier in the magic shop, falls from one single point. Rain would not seem to stand still, but flicker all around the place. However, Atlas clearly states at the end of his trick that it was done using strobe lights and rain machines, therefore the rain was not real rain and the machines could have been calibrated to be consistent enough to achieve the desired effect with linear streams of "raindrops".
When Dylan and Thaddeus arrive at the magic shop and speak to the grandmother, Thaddeus translates her response. Dylan replies that if Thaddeus's Mandarin was good, he would tell him. However, Macau Chinese - like Hong Kong - speak Cantonese, not Mandarin.
When Rhodes is in the safe at the bottom of the river, water rushes in. But in the shot from outside the safe, no air is escaping.
When Rhodes (Mark Ruffalo) goes to the prison to see Thaddeus Bradley (Morgan Freeman), he uses a different agent's ID to get in. Bradley tells Rhodes, since he was the arresting officer, he can get him out of prison for 24 hours using by completing a form that's by his printer... which would be his own name, not the name he used to get in the prison to see Thaddeus.
The emergency exit sign in the private jet in the end is for buildings not aircraft.
Since the climax of the film takes place on New Years Eve, the beginning must be happening sometime in December, yet the establishing aerial shot of New York City includes trees on Liberty Island and other islands which are lush with green leaves.
When the horsemen are being thrown out of the plane, the door is on the right (starboard) side of the aircraft. When the final reveal is shown, the door is on the left (port) side of the aircraft.
In the first New Year's Eve scene in London, dozens, perhaps scores of people have near identical umbrellas. In real life such a diverse population of random visitors would have a hugely diverse selection of umbrellas.
The scene in which the Horsemen are being searched by security for the Stick, Merritt has the card looking who to throw it to; we hear Lula saying "I do not speak the language." We can clearly see her turn her head to Merritt and her mouth isn't moving while the line is being said.
The remote controlled camera is visible in the scene where the truck full of the horsemen is pulling into the garage. Look as the camera pans down and we see the top of the van. The camera looks like a box attached to some poles.
When the horsemen are broadcasting around the world, they cut to Sydney and New York where it is seen as being night time. In reality it would be day time and early evening.
During the London sequences, the FBI maps' GPS locators indicate that the Horsemen are in Limehouse, when in fact they are in Greenwich.
When the new safe is dropped in the ocean, the skyline in the background is Dallas TX.
When Merritt is on his way to perform his solo show near the end of the movie, he bumps into his brother Chase, later, Chase bumps into Jack and thinks he is a hobo, when then Jack hypnotizes him, yet meanwhile Jack should be performing his "find the lady" trick somewhere else.
The watch made by his father can never have uvLED inside, that technology was not available at that time.
Prior to the final show, when the horsemen are performing separately at different locations in London. [Scene X] Jack is busy with his card swap trick. While others are doing their tricks too, [Scene Y] McKinney is believed to be tricked once again by his face sharing brother at another location... Dylan gets a call and then he calls everyone to wrap up and meet somewhere. [Scene Z] Jack also wraps up and leaves the place... OK. Later after the expose in final show McKinney tells the story about how Jack successfully hypnotized his brother [in the Scene Y]. How could Jack be at two different locations at the same time? He was performing somewhere else when he did hypnotism at another location and then wraps up his trick back where he started.
Thaddeus was behind Dylan's receiving his father's watch in the magic shop in Macau, although there was no way for Thaddeus to know that Walter will decide later to lock Dylan in the same safe and throw him in the river.