When Mengele tears off Mark's face, he removes much of his front hairline, while the incision was only made at the top of his forehead.
Early into the movie when Paige and Mark are drilling into the Antarctic floor, between shots, the drill Paige is holding changes appearance constantly.
Dr. Moss catches the sample container after it is dropped by his associate carrying the tray, but he catches it at a much higher level than when it was dropped.
When dude rips his friend's transplanted face off the Nazi, the forehead cut line is even loser then it was when removed from his friend, or transplanted onto the Nazi.
When Paige enters the freezer containing body parts, the door behind her clearly remains open, but in the next shot it slams shut.
As the robot first appears outdoors, the flag in the background depicts a counterclockwise-pointing swastika, as though the image has been revered.
Mengele would not have been able to shoot an Allied soldier in the eye using a pistol while holding on to a taxiing plane from hundreds of metres away.
When the team members rappel into the crevasse, the are shown upside down. One does not rappel upside down.
The swastikas on the 'Vril' (the flying saucer) do not match -- one is reversed.
The dialogue mentions Josef Mengele's death in the 1970s. The real Mengele did die in 1979, but in the movie, Mengele's been hiding at the center of the Earth since 1945, so he's logically been presumed dead since then and there was absolutely no reason for him to fake his own death 3 decades later.