Revealing mistake: After Wonder Woman kills the final terrorist with a huge blast, debris rains down on the police below. Not only are none of the officers injured, the debris doesn't cause any damage to the vehicles whatsoever. There are no broken windows, broken windshields, broken light bars, or any dents and scratches despite massive chunks of the building falling all over the vehicles.
Other mistake: After realising they can track the mother box, Batman says they have to go back to his base to use one of his satellites to track it, and they all do so. Which completely ignores the fact that Cyborg could easily access the satellite feed directly using his abilities and display it to the others on any nearby screen in Star Labs.
Other mistake: Barry comes back to his home, replaces the fuse in the fuse box, flips the handle (begs the question of why remove the fuse AND turn off the switch, but whatever), and within 2 seconds every screen in the place is lit up. Sure, a TV might come on that fast, but plenty of those screens are showing data, analysis, etc, running off computers, which would take longer than that to boot up and get everything running.
Other mistake: Apparently, one of Aquaman's superpowers must be to be super-heavy, since surfing on the body of a parademon he lands on a five story building and that somehow slices the whole building in half, complete with explosions that leave him unscathed and don't turn the parademon's carcass into a pulp. (03:15:15)
Factual error: In Part Two, as Diana explains to Bruce Wayne the history of the Mother Boxes on Earth, we see an extended flashback of Earthly gods and warriors in epic battle against Darkseid. When Diana says, "A golden age of heroes fighting together," we see a closeup of an Amazon archer drawing back an arrow right-handed, leaning right, and releasing it. However, the arrow is UNSUPPORTED on the bow, so she couldn't possibly aim or control the arrow. A truly amateur mistake. (01:03:59)
Plot hole: WW mentions that "As Darkseid waged war on Earth, he found a secret there", that being the Anti-life equation. But later on it turns out that after being defeated, planet Earth is so "anonymous among a trillion worlds" that he never manages to find it again and destroys another 100,000 worlds (his words!) to look for it again. That would mean that they lack any sort of navigation, and it's hardly possible anyway that the planet would be "anonymous" when it contains what Darkseid wants the most.
Continuity mistake: Hippolyta is running away from the chamber of the Box and screams to ready the hammers. Notice how she is running in the middle of the corridor and if anything her steps bring her closer and closer to the line between the stones running exactly in the middle. A parademon bludgeons the other amazon that is following Hippolyta, and you could see him in the background earlier too preparing to do so, but now the queen is running towards the right side of the entrance, leaving room for the camera. (00:29:45)
Continuity mistake: When Flash talks to his dad at the end of the movie, he puts a paper against the security glass. The paper is tilted towards his right a bit, but in the reverse shot it's tilted more heavily and in the opposite direction, and so it stays for the rest of the scene. (03:37:05)