Character mistake: When Morpheus is explaining to Neo the reason the Matrix exists, he states that a human being gives off 25,000 BTUs of body heat to help power it - this is just not true. Even human beings that are running a marathon can't give off that much heat, and the humans in the Matrix are all lying down, not moving.
Character mistake: Why is Electra using Windows 2000 (still in beta at the time) to survey her oil empire? (I think it's just after she's asked M to come and help her) You shouldn't use beta software for anything important....
Character mistake: When Gordon describes Leonardo Da Vinci's idea for a flying machine, he mentions Da Vinci designed it in 1540. However, designs for Da Vinci's flying machines were dated as early as 1488 and Leonardo Da Vinci died in 1519.
Character mistake: When a helicopter winch operator tries to get the gurney with injured Jim inside the helicopter, he doesn't have any gloves on. No wonder his hand gets injured when the cable breaks loose.
Character mistake: Henderson asks the guy as he's leaving the room to get him some bourbon. They already drank in the bar and spat it out when they found out that everything had turned to blood, so it would still be blood.