Revealing mistake: In the 100 Agent Smith fight, you can see (especially in the larger shots) that there are many Smiths that don't look like Smith at all. Also, for the ones that have Smiths face digitally painted on them, when they turn around you can see that the backs of their heads are all different.
Revealing mistake: After Neo takes the pole for the fight with the Smiths, we have some "bullet-time" shots of Neo fighting with the pole. In those shots we can see that Neo and the pole are computer made. This becomes even more obvious because before using the pole Neo was crushed into a wall and got all dirty and the computer-made Neo is completely clean.
Revealing mistake: After Neo has knocked out the first two agents after the meeting scene, he kicks the last one into the pole of a streetlight. When the agent falls to the ground, he moves his arms from a position where he could brace his fall to one where he could protect his face from the shattered glass that comes directly after when the glass from the streetlight falls and breaks in front of him.
Revealing mistake: When Neo fights the Agents near the beginning, he knocks out the first by knocking him into the wall. He then knocks the second into the wall by spinning round on the third Agent's back. After the shot of the second Agent hitting the wall both Neo and the Agent go from standing completely still displaying a clear "Action" cue.
Revealing mistake: In the scene where Neo is fighting the 100 Smiths, there's a shot of Neo baseball-smacking a Smith into a set of windows. While the scene is a bit blurry, it's pretty apparent that the Smith literally falls apart when he hits the window, like he was made of sausage links.
Revealing mistake: In the highway chase scene, Trinity, Morpheus, and the Keymaker pull off onto an exit and the twins follow. If you look closely at the exit and the overpass, you can see that it ends after it goes over the highway. The exit/overpass goes nowhere. You can see this during the various shots when Morpheus faces off against the twins and later when Trinity looks back at the overpass from on top of the Ducati carrier.
Revealing mistake: During the freeway chase there is a shot over the shoulder of one of the twins as he shoots at our hero's car. He rakes his stream of bullets from left to right following the car as it passes behind a white truck - notice, however, that there are already bullet holes in the truck before the stream of bullets ever reaches it.
Revealing mistake: Just as Neo begins fighting in the "Burly Brawl," take a look at the Smith to the right of him (he puts his hand on Neo's shoulder I think). The face is not Hugo Weaving's but the face of the stunt person. Looks like the effects crew forgot to put Weaving's face on that guy.
Revealing mistake: After the twins shoot the black Oldsmobile, it hits a barrier and does a spiral, and lands on its roof. That's impossible because the car's fender would have been torn off. It would have skidded a sharp right and went under the semi that Trinity was driving by. That indicates that there was a ramp to lift the car off the ground and do the spiral.
Revealing mistake: In the freeway chase scene, look carefully at the interior of the cars that get flipped over by the Agents and the Twins. All of them are completely empty, meaning that either they are remotely controlled or they are simply towed by a hidden vehicle. This is most obvious when they are flipping in slow-motion.