Audio problem: In the climactic scene where the 747 is taking off, the general applies full power on the throttles and the engines spool up to full power as they should. However, in the next shot from the cockpit, the general is pulling back on the yoke, nobody's hands are on the throttles at all, and yet we hear a second (apparently) identical increase in engine power. Pulling back on the yoke has nothing at all to do with the engines - the noise we hear should remain at a constant level. (01:50:15)
Audio problem: When the terrorist at the church goes inside and asks "where's Cochrane", the other replies "he didn't make it." His mouth isn't moving.
Audio problem: When General Esperanza is trying to land the plane there's a hell of a lot of noise from the wind blasting in through the shattered windscreen. Yet when we see McClane listening in to his radio communication he can be heard completely without a trace of wind noise.
Audio problem: When McClane goes to Lorenzo's office for the first time, we hear Lorenzo say "who is it? Come in." His lips don't move (which you can tell despite his mug being up to his mouth).
Audio problem: John is on the wing of the plane, questioning why he is there. His mouth isn't moving with what he is saying.
Answer: In the first movie he's a New York cop visiting his wife. In the second Die Hard, he tells the airport officer that he's LAPD and moved there because of his wife's job. In the third Die Hard film, he most likely went back to New York because of marital problems and became a New York cop again.