Deliberate mistake: When John crashes the car against the elevator room, the cables he and the woman hold on to are clean, when they should be greasy - thus allowing for them to hold safely.
Deliberate mistake: While McClane drives in the tunnel, Gabriel orders, "Open the gates on the north side entrance." A few shots later the signal light changes from red to green and the north gate opens. The "warning" sign is backwards in this flipped shot. Then, when Gabriel opens the gates at the south entrance, there's a proper shot of the "south" gate opening. The two shots of the "north" and "south" gates opening are the same, but one is reversed. All the dirty smudges on the signs, booths, etc, are identical. (00:41:20 - 00:42:05)
Answer: The radio line mentioned the officer's name in the seconds after Gabriel's voice recognition picked up Farrell's name.