Revealing mistake: During the battle at Troy, when Odysseus and Achilles are standing together surrounded by Trojan warriors, Odysseus lashes out with his sword and kills a warrior (just before Hector appears and challenges Achilles). If you look carefully you'll see that Odysseus merely swung his sword for show; it didn't connect with the Trojan warrior who fell dead.
Revealing mistake: The gaps between the boards of the Trojan horse are much wider in the inside than on the outside. This reveals that the soldiers are not really inside the horse, but on a set.
Revealing mistake: Achilles takes the spear that he later throws at Hector out of a dead person's body, but the spear has no blood on it at all.
Revealing mistake: After the sack of winds is opened, Eurylochus is shown grabbing onto a rope in midair, and the string holding him up is visible.
Revealing mistake: After Achilles dies and the Greeks are standing at his burial, Achilles is breathing for the first few seconds of the shot.
Revealing mistake: It's stated that Achilles has been killed. He then is shown lying face up on his pyre. But if you look closely enough (zooming in might help) you can see that he's still breathing.