Deliberate mistake: Leaven spends too much time thinking about if 645 and 372 are prime numbers to be believable. But she knows the factors of 649 and the result of 26 to the power of 3 almost immediately. Plus anyone with any maths knowledge whatsoever would know that any even number (or one divisible by 5) can't be prime, so 645 and 372 could be dismissed without even thinking about it.
Deliberate mistake: When the ship crashes into the dock, all of the scenes leading up to the crash and after it show a clear view across the harbor as we can see lights and land on the other side. It's not until the ship is about to crash that a thick and dense fog appears and disappears within a minute to mask the ship to create a more dramatic effect.
Deliberate mistake: The slow rate of the radio telescopes at the VLA has been greatly increased for the movie. They are actually quite slow and make little to no noise.
Deliberate mistake: At the end of the movie Rocketman shows his lady friend a space blanket, out of which he states he made something: ballroom dancing outfits. The difference in the material is just too obvious for this not to be a deliberate mistake. The outfits aren't the slightest bit related to the foil they were supposed to be made of.
Deliberate mistake: In the finale when Kadeem Hardison is beat up by the motorcyclists he performs a back somersault. As he falls to the floor, you see a white guy's back, despite the character being black.
Deliberate mistake: The shot of Orbison's house buring at the end is quite obviously fake. You can see that it's made of cardboard and paper, and even the windows burn and curl like normal bits of paper do when set alight.
Deliberate mistake: When they fight in the hangar and Troy is thrown back by the jet engine, his face and other parts of his body should have been burnt quite heavily. Such a jet engine has temperatures higher than 450°C about 5 meters away, and Troy is even closer than that.
Deliberate mistake: In the very first shot of Korben Dallas, his bedside clock is facing away from him, towards us. We can read it, but he shouldn't be able to because he'd be looking at it backwards.