Continuity mistake: When Obi-Wan, Qui-Gon, and Darth Maul get cut off from each other in the corridor and Qui-Gon kneels down, Obi-Wan is standing in the background with his lightsaber pointed downwards. In the next shot, it's pointing upwards. (01:51:45)
Other mistake: When the Jedi, the Queen and her entourage enter the hangar before they leave Naboo, only four of them are reflected in the shiny starship above them. The others are not. (00:23:15)
Revealing mistake: When Darth Maul is harassing Obi-Wan as he dangles in the pit, the first swing of Darth Maul's sabre is out of sync with the resultant sparks (very evident in slo-mo). (01:58:50)
Revealing mistake: In the final celebratory parade/ceremony scene, sun/shadow angles vary drastically between shots and a variety of items (flags for example) cast no shadows at all.
Continuity mistake: At the start of the lightsaber fight in the hanger all three go through some large doors and into the big generator room. Behind them a set of stairs have appeared.
Continuity mistake: When Obi-Wan runs towards Darth Maul and Qui-Gon he is halfway down the walkway, but when it cuts to Darth Maul and Qui-Gon Obi-Wan is now at the far end.
Audio problem: When the droid army are marching into battle at the end we hear a marching army, but they're walking on grass and the sound should be muffled in some way.
Revealing mistake: In a few of the wider crowd shots during the pod-race, you can make out some oddities. In particular, if you look closely, you'll see the same group people copy/pasted over and over in different places and different configurations in the audience to make the crowds seem bigger. (Slow motion or pausing helps, but is not necessary.) This is actually a relatively common special effects trick to make crowds seem bigger.
Continuity mistake: When Qui-Gon and Obi-Wan are fighting battle droids in the Trade Federation ship, Nute Gunray says, "I want droidekas up here at once." As he says this, he turns his head, causing the "straps" hanging from each side of his headgear to hang slightly behind his shoulders. In the next shot from behind, they are hanging in front of his shoulders like they normally do. (00:05:55)