The time-stamp on the footage of the un-mutated turtles in the lab says 1999, which matches the chronology of the film. When we are shown the young April taking this footage in a flashback, she is taking the footage on a bluetooth-enabled digital camcorder, which would not have been available then. The camcorder is also a MicroMV camcorder. MicroMV was not invented until 2001. A MiniDV, Digital8, or even an analog camcorder would have been more period-accurate.
Near the end, when the tower is falling, Raphael gives a speech about being brothers. However, if you look in the background, you can see that the scene is moving in slow motion, but we are hearing Raphael speak at regular speed. In reality is Raphael speaking really quickly?
When April uses her phone to make a Skype video call to the dock worker (indicated by the Skype logo on the screen) the worker asks April how she got his number. Skype video chat uses screen-names, not phone numbers.
When looking through her father's journals, April sees the Japanese characters for "family" that is noted as the Turtles' calling card. This seems like an unlikely thing to put in a science journal, especially considering that the Turtles learned all they know of Japanese culture from a book Splinter found possibly years after they were put in the sewer.
Consistent with TMNT lore, Raphael holds his sais with two fingers over the "yoku", or curved part of the sai. No one who is even remotely trained in sai-jutsu, or using sais in combat, would hold them in this fashion as it defeats the purpose of the weapon.
During a flashback scene, young Raphael carves a turtle statue out of a tree stump using his sais. This is highly unlikely since sais are truncheon weapons and are not sharp, but used like batons for bludgeoning.
In the final showdown with Shredder, Leonardo throws one of his katanas at Shredder, which falls to the ground with Shredder. However, throughout the rest of the scene Leo still has two katanas on his back.
Near the beginning when April interviews the fitness trainer and is jumping on the trampoline she is wearing a pink shirt and a black coat. When she walks back to the news van she is wearing a gray shirt with a burgundy coat.
At the end of the movie when Michelangelo accidentally blows up Vern's new car he fires the missile at the rear of the car. The car flips end over end forward, but when it lands the rear end of the car is still facing them. Note the tail lights and open trunk.
During the snow scene Michelangelo's rocket skateboard randomly appears and disappears. In fact, he didn't even appear to have it when the turtles were taken from the sewer, but had it when on the mountain.
April is Skyping a detective and he asked how she got his number. When you use Skype to call numbers you don't get a video chat. The only she could video chat like that was to call him on Skype and he would have to be on her skype contacts to do that.
(at around 26 mins) During one of April's video tape reviews one of the turtles' tanks is labeled "Michaelangelo" which is incorrectly spelled. It should be "Michelangelo". This may be explained by the possible lack of knowledge on the part of the very young April.
When April O'Neil calls Vernon Fenwick (who drives the channel 6 van), he answers his phone and immediately says "O'Neil". When the scene cuts back to O'Neil, we see she called from a payphone. Fenwick would have no way to know it was her calling.
If the mutagen in the turtles' blood was so precious that the Foot needed to take three turtles, why not take Splinter too? After all, his blood has mutagen in it too, and taking three turtles rather than one suggests every ounce of mutagen is valuable.
Consistent with TMNT lore, Splinter and Shredder seem to have an ongoing rivalry with each other. In this incarnation, their epic battle in the sewer is actually the first time they meet each other, thus rendering any history they may have as nonexistent.
April tries to sell the "Vigilante Turtles" story to her boss who doesn't believe a thing and ends up firing April. Minutes later April shows Eric Sacks a photo of the turtles. If she was so interested in getting the story and she already had a photograph, why didn't she showed the photo to her boss?
When April approaches her editor (Whoopi Goldberg) about the "vigilante story", only to be shot down; it's because April tells her there's "nothing left behind". Despite the fact that April hadn't shown her editor the photo of the Turtles, she had shared the photo of the symbol for "family" and said that the vigilantes had thrown shipping containers at Foot Clan members (which was seen on-camera) -- plenty more than "nothing".
When April is on the docks after the initial attack the turtles make and thwart the foot soldiers - April takes a photo of the "family" insignia on the container with her phone and for a brief moment during the camera angle change you can see that the forward-facing camera is in fact on, thus making it a "selfie".
When Shredder assaults the Turtles hideout where Raphael is leaning over Splinter, the scar on Raphael's lip changes from left to right, as does the hole in his sunglasses changes from right to left lens, indicating that this shot was reversed.
In one of the scenes with the turtles in the stasis tanks Leonardo is shown in a reverse shot where his shoulder strap is over his left shoulder instead of his right shoulder. Interestingly, though, the animators did digitally reverse the NYC button on the strap so that it was not reversed.
At the beginning of the movie April is doing a story about how to stay fit as it is the first day of Spring yet at the end of the movie the Sachs antenna with the toxin inside falls and destroys the Tines Square New Year's Eve ball and 2014 sign. How can it be the first day of Spring yet the New Year's Eve ball is still on display.
At the end of the movie during the beginning of the fight scene atop the Sachs tower you can clearly see that One World Trade Center is just a few blocks away but then during the remaining of the fight we see that the Sachs Tower is back in midtown and One World Trade can be seen way in the distance.
Shredder has Raphael under his foot at one stage, his shell is cracked and he is about to be killed. Shredder just walks away for no apparent reason. Then later he is on top of the tower working a computer to release the toxin yet we're given no explanation as to how he has gotten there so quickly (or why he would be doing this himself or how he knows how to work a computer that apparently is quite complicated since even Donatello has quite a lot of trouble working it), Sacks took a helicopter to get there and the Turtles basically fell down a mountain at high speed yet somehow Shredder gets there before them.