- Click to focus game
- Press enter to chat
- Type a few bogus chat messages so you can see the chat area
- Open up the chat box again
- Fire at some enemies
- Now, the chat messages have faded out, but the chat box remains onscreen (although you can't type in it)

You probably want to intrinsically tie the visibility of the chat-input to it's focus state. It's pretty easy to do this with css, but any way works really.