The film takes place in 2012. At the beginning of the film, Heather Miller is shown to have been a baby in 1973. This would make Heather at least 39 years old. However her character is obviously much younger than that (her actress Alexandra Daddario was only 26 years old at the time of filming).
The characters of Sheriff Hooper and Burt Hartmann were present during the 1974 massacre of the Sawyer family where Heather was adopted. 39 years later and these characters don't appear to have aged ever since.
During all scenes of the VW camper van driving around on the grounds of the house there is tyre squeal this would indicate rubber on tarmac/concrete although all the drive consists of is dry dirt track which would make no noise at all.
Despite the fact that Leatherface hunted down Heather and her friends relentlessly to kill them, once Heather discovers they are family she immediately warms up to him and begins fighting by his side against the vengeful police department. This is just not realistic in any way no matter how you look at it.
In the Sawyer family police files that Heather goes through at the station, they state that Drayton Sawyer is the father of Loretta, Heather's mother, making him Heather's grandfather and making his son Leatherface her uncle, not her cousin, as he is repeatedly called in the movie.
Drayton Saywer is repeatedly referred to as Leatherface's father. In the original two "Texas Chainsaw Massacre" films, fans often mistook the Cook, Drayton Sawyer, for the father of the Hitchhiker, Leatherface, and Chop-Top. However, Tobe Hooper has said that they are all brothers and, although not explicitly stated in the first movie, Drayton even refers to them as his younger brothers, not his sons, in the second one. This is either a notable change or a mistake in "Texas Chainsaw 3D", depending on how you look at it.
In the letter left by Verna Carson she tells Heather that Leatherface comes with the inheritance and if she takes care of him he will do the same as she is his only family left. It seems pretty illogical that she would set her up this way for the chainsaw wielding maniac that has no idea that his cousin is still alive with just a measly letter.
As revealed in the end of the film, the letter left by Verna reveals Heather's birth name is Edith Rose Sawyer. When Leatherface finds Heather tied up he notices the family sigil branded on her collar bone and in response Heather tells him "It's me! Heather!". This is an illogical response as Leatherface does not know her by her adopted name but rather by her birth name.
The date that the farmhouse was burned and the family was killed was August 19th. However, as seen later in the film when Heather is looking at the newspaper, the story was on the front page of the paper dated August 19th. The event would have occurred after the newspaper was printed and distributed.
When Kenny is dragged into the basement he drags the carpet on the stairs down with him. After Heather escapes from the basement a few minutes later the carpet is back on the stairs when she runs up them
Leatherface appears to move very fast and limber and very fit to be able to chase around Heather and her friends. Given that the film is set 39 years later he would be in his 60's and not physically fit to terrorize the characters as seen in the film.
Heather works as a butcher in a supermarket and is seen wearing a beanie while cutting slices of meat. As per supermarket policy Heather would be required to wear a plastic cap in her hair so that her hair does not fall into the meat.
Nikki ends up seducing Ryan to have sex which effectively means he cheated on Heather. Since they both die immediately afterwards Heather never finds out. This is a subplot that had no purpose whatsoever in the film and is basically filler material.
When a character uses FaceTime to show other characters what he sees in the Sawyer house, his iPhone clearly focuses (indicated by the blue square in the center of the screen) which does not happen during a video call, indicating that he was only on his camera, and switched to it every time he was re shown.
While Leatherface hangs up Kenny on a meat hook he begins to retaliate and Leatherface responds by bisecting him at the waist with a chainsaw. During this entire sequence Kenny is thrashing in pain in shots of his upper torso while his lower torso remains still revealing the use of a dummy.