How do you handle role-players who like to threaten to leave a game if events in the game do not go in the direction they had in mind?
Let them leave. Blackmail, if allowed once, will always come out again. It's not worth the stress and possible bad blood with the other players to keep someone with that kind of attitude in a game.
If I think they were just being self-absorbed then I would gesture to hold the door for them while they go out as it would be better for everyone if they did leave.
I would not display any emotion. I would just say, "Ok, nice playing with you… "and then turn my head back to the game and carry on.
If anyone tries to do this you should let them leave and be cautious if he/she wants to play again.
I haven't seen this happening many times while I was playing.
I like what reader said. These kinds of players, or non-players depending on how you see it, feed off of emotion trying to get everyone to care about their wanting to leave.
Don't let the door hit you where evolution split you. I never negotiate with terrorists. If you aren't having fun and want to ruin the game for others with your drama then leave. I don't tolerate drama and trying to bully me into changing my decisions or hit rewind because your dice betrayed you.
Tell those players that it's "Tough luck, but I am not going to cater to your wishes just because you'll leave if I don't. There's no solid reasoning behind your statement, and if you want to leave, then leave. I am not stopping you, just as you are not stopping what is about to happen."
If it's about set boundaries and a Dungeon Master going past those anyway, then yes, I'd give full credits to the player, but in this case it just sounds like a demanding child that wants the game to go its way.