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: 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.