Factual error: When Neo was getting shot outside room 303, Smith fires 10 rounds into him. More if you count every time Neo is shaking in the seat as additional shots. The Desert Eagles in the movie are chambered for .50 AE. You can tell this because the barrel isn't fluted. This means it has a 7 round magazine. Agent Smith runs out of ammo in the subway, so it's not like he can hack his gun because of his software.
Continuity mistake: When the agent is running after Trinity, Trinity gets to the end of the roof and runs on. Notice that the end of the roof is red, or a rusty like colour, but when when the agent starts shooting at her, it's turned from red back to probably its original colour of silver. (00:04:15)
Visible crew/equipment: When Smith and the other Agents turn round in the corridor at the end the white reflective screen is visible in Smith's left lens as he is turning. (01:59:40)
Continuity mistake: When Neo steals the guy's mobile phone there is a shot looking back at the man (agent Smith) waving his arm . On the pavement there a few people going both ways. When the close up of Smith happens there are loads of people going left to right and more people on the right side of the screen. (01:54:28)
Continuity mistake: When Neo and Morpheus are inside the hotel room and Morpheus is giving the offer to Neo, the reflection in either part of his shades shows that he is holding his hands away from each other. Yet, when Neo goes to pick up the red pill, they are close together, then we see them further apart from the reflection inside the shades. (00:28:15)
Factual error: In the "Deja Vu" sequence, when Mouse opens the box containing his Auto Shotguns, they are stored with a few belts of .50-caliber BMG ammunition, which would be useless in a shotgun. Also, despite firing about 20 or more 12-gauge shotgun blasts at the SWATs, he fails to hit anyone, even though that equates to about 240 flying lead pellets.
Visible crew/equipment: When Neo and Morpheus are in the training program with the girl in the red dress, after Morpheus has paused the program you can see a camera and a black fabric cover around it reflected off Morpheus' sunglasses for a considerable amount of time. Noticeably in the shot considerably close to his face while talking. (00:58:00 - 00:59:00)
Other mistake: They are presumably in the US - trying to make it look that way, anyway. Cars driving on the right, speaking English etc. However the filming location was overseas, revealed by the elevator in the lobby scene. When they hit the button, the plate refers to the elevator as a lift.