Revealing mistake: Qui-Gon's "comlink" is actually a Gillette Sensor Excel woman's razor. Borderline trivia, but it's such a recognisable shape with minimal effort to hide it. (00:50:45)
Revealing mistake: After everyone leaves Tatooine after the short stay they land on Coruscant in the middle of busy air traffic. Unfortunately, the ship that they fly in (the Nubian), with its incredibly shiny and mirror-like appearance does not reflect the bustling traffic above, nor the people walking out of it. It only seems to reflect the basic landscape. (01:17:35)
Revealing mistake: When Qui-Gon is talking to Jar Jar for the first time, they are walking through the forest. A branch with leaves comes up and Jar Jar dips his head to avoid it, but his eye and part of his head go through the leaves. (00:11:30)
Revealing mistake: When the Queen blasts the window to gain entrance into the upper floor of the palace, a shot from down the corridor reveals that the window was already open before the sparks and smoke appeared on screen.
Revealing mistake: When Qui-Gon is attacked by Darth Maul for the first time, he escapes by leaping onto his ship which quickly takes off. As they zoom up into space, he discusses the mystery attacker with Obi-Wan. However, the lighting from outside the windows never changes giving the impression that they are static on a set rather than whizzing up into space.
Revealing mistake: When the Queen is escorted down the stairs in the Theed Royal Palace, right before Nute Gunray says, "Process them.", watch the space between Sabé (the Queen's decoy) and Captain Panaka, before a handmaiden and one of the Queen's guards block the view; the battle droid behind Sabé doesn't cast a shadow on the railing, as everyone else does in this shot. (00:21:00)
Revealing mistake: When Qui-Gon and Obi-Wan are being gassed, Nute Gunray's hologram says, "Destroy what's left of them." After he says this, watch the battle droid on the left side of the screen closest to the hologram. One of its fingers goes through the barrel of the gun it's holding. (00:05:15)
Revealing mistake: During the second lap of the Podrace, the announcer says, "Skywalker's spinning out of control" and there's a shot of the audience. After Anakin has won the race, the same shot is used again. This can be seen, for example, from that the people on the steps make the exact same movements in each shot. (01:01:50 - 01:06:50)
Revealing mistake: When the Gungan army set up their shield at the Naboo plains, you see the Trade Federation tanks coming over the hill. When the camera turns to behind the tanks, you see the bottom of one of the beige tanks disappear slowly.
Revealing mistake: When Jar Jar tries to retrieve the wrench that he dropped in the engine of Anakin's podracer, the metal slat above his hand bends like rubber. (00:46:45)
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.
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.