When Max is about to leave his apartment in the morning, he shuts off the stove after a kettle starts whistling. The steaming item on the stove is an espresso pot, which does not whistle. A kettle is on the back burner, but it's not being used.
When Mona Sax arrives at Aesir Pharmaceuticals, she shoots through a door, near the lock handle, to open it. When she kicks the door open, the door handle, lock, and deadbolt are undamaged, and the deadbolt is not locked.
When Max stands in front of the two lamps on top of the Aesir building, he drops his pistol and kneels down; the gun falls in the yellow area with the grip towards the viewer. Then the pistol is laying more behind him in the black area and with the grip towards the building.
The film is set in New York. Yet when Max asks Alex "where are we going?" in Alex's car, a circular blue, white, and red sign is visible on a building in the background. It's a sign for the CAA (Canadian Automobile Association). A Canada Post delivery truck is parked in front of it. Alex's car also passes a TTC (Toronto Transit Commission) bus for the disabled.
(at around 46 mins) Max is driving through the city (supposedly New York City) when behind him through the windows of the car, there is a "Tim Horton's" neon sign. There are no Tim Horton's in NYC (the nearest is in Syracuse, according to the company website), but they're all over the place in Toronto (where the film was shot.)
Jim Bravura has a detective's badge, and is frequently called a detective. Yet the door to his office and the name plate on his desk identify him as a lieutenant. In the NYPD, he would be called a lieutenant, even if he was assigned to a detective squad.
In the Roscoe Street Station bathroom, Max fires a revolver 6 or 7 times (an echoing boom is heard when the mugger is running on the tracks). Max is using a Taurus Judge revolver, which holds 5 bullets.
When Gunnery Sergeant. Jack Lupino is shown on video during his interview, his Gunnery Sergeant lapel pins are pointing in the wrong direction. The top should be placed vertically, with the single point up and the center of the insignia on a line bisecting the angle of the collar's point. Also, he is referred to as "Sergeant", as a Marine he would be addressed as "Gunnery Sergeant".
When Max Payne shoots out a couple of fire sprinklers, all the sprinklers in the floor go off. Each sprinkler is designed to be activated only if the temperature under that sprinkler melts its fusible link or if the sprinkler head is physically damaged. All the other sprinklers would not be activated unless a very large fire set them off by melting their fusible links.
When the Marine is being interviewed about the drug's effects, the rank on his collar is upside down, and he seems to have been promoted. The rank insignia are not USMC insignia. USMC insignia would have crossed rifles between the chevrons (pointed end at the top) and the rockers (the curved end at the bottom). The rank shown is the army Sergeant First Class (E7), and in the army he might be called a sergeant. In the Marine Corps the rank of E7 is a Gunnery Sergeant, and would be called Gunnery Sergeant or more informally Gunny.
Just before the fight with Lupino there's a guy coming behind and above Max and there's a slow-motion shoot out. The first shot comes almost in front of Max but the rest of them seem to hit the drugs on a table that's at least 2-3m on Max's right side.
At the shipyard, when BB is about to execute Max Payne, Max jumps into the water. BB and his partner shoot at him, hitting Max in the left leg. Max surfaces, takes the drug, stands up in a fit of rage, then limps away on the wrong leg. The bullet hole is now in his right leg.