Audio problem: When Freedom shuttle is being destroyed by the asteroid field, you can hear Harry say, "Sharp, what the hell is that? Is that the Independence?" but you can tell by Bruce Willis' mouth that he isn't saying any of those words.
Audio problem: During the sling shot around the moon we hear a sound in more or less every shot of jet engines being revved up. They couldn't rev the engines up several times - it's not like such engines have gears.
Audio problem: When the Russian space station is on fire, and the crews are running out, you can hear Harry say, "Where's the rest of the team?" but his lips don't move.
Answer: If you look at the deleted scenes on the special edition DVD, you will see a deleted scene in which A.J. asks what they needed a gun for, and Max explains that it's for debris elimination, in order to take out small rocks in the way.