When Bane is talking to Neo while holding the knife to Trinity's throat, the blood appears and disappears on her throat.
When Neo bids goodbye to Morpheus the plugs on the back of Neo's neck keep disappearing and reappearing.
When Trinity, Morpheus, and Seraph go to visit the Merovingian, the Merovingian is seen eating olives from his drink (a Martini). At first glance, there are only two olives. He takes half of a bite out of the first olive, then finishes it, then eats the second. However, just moments after eating the last olive, a third olive magically appears on the toothpick.
As the Hammer is reaching the gate to Zion, the hovercraft's gravity pads (glowing pods) switch between being damaged and fully functional. In addition, all of the aft pads were damaged at one point and the hovercraft was seen dragging the tail end on the tunnel. However, once the ship reached the gate, all pads, including the damaged rear pads, were shown to be fully functional.
When Neo is talking to the Oracle the magnetic letters on the refrigerator behind him disappear then reappear.
In the train station scene where Neo is talking to Rama Kandra they talk about how Neo has never heard a machine talk about love when in-fact Persephone spoke about love several times and wanted to experience it again by kissing Neo in The Matrix Reloaded.
When Trinity is shown dying, Neo bends over her and, in doing so, slightly moves her body which, in turn, moves the fake poles that are supposed to have gone through her body.
The presumably metal pipes that Trinity is impaled on and that seem to pin her to the ground, wobble and bend, revealing them to be lightweight props.
Obvious stunt double when Neo is punched into the wall by the train man.
In the final battle in the crater, after Smith picks up Neo and punches him, you can see a wire just above Neo's right shoulder dragging him along the ground.
When the Train Man is running from Morpheus, Trinity and Serif, the first shot of the upcoming train where he makes his escape shows no rails in the track bed.
The underground area Zion is located in is full of tunnels, perfectly usable for travel, as shown in all three movies. The machines, with their advantage in numbers, shouldn't need to dig down to the city, they could access it through the tunnel system by breaking through the gate.
When Neo is talking with the Oracle, he says "How did I stop four sentinels just by thinking it?", when in fact, at the end of The Matrix Reloaded (2003), we saw that there were actually five sentinels that Neo stopped, not four.
Even though Neo planned to peacefully surrender to the machines when he reached the city, he made no effort to communicate this to the machines via radio. Considering that Neo still had the 'codes' that the Architect wanted, and that the EMP from a single ship would barely make a dent in the city's defenses, the machines would have surely been open to a peaceful surrender, but Neo does not even attempt this, resulting in Trinity's death.
Even though Commander Lock stressed the importance of destroying the digging machines earlier in the film, no effort is made to do so when the dock is (temporarily) free of sentinels, and he seems only interested in disciplining the crew of the ship that disabled their equipment.
Zion ships are able to have EMP's but Zion itself fails to have any around the walls where they know the sentinels are coming from, in fact, the humans have no EMP grenades or EMP guns. If they have the tech to put them on a ship, they could be everywhere else, essentially making Zion impenetrable by the sentinels.
Neo forgets that he now has powers outside the Matrix when he fights Bane, until his new abilities allow him to see without eyes.