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: 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.