As far as I know, the arcade itself, or rather the games, do not log you out. It's possible that your set up to be logged out of TCS after a certain time of what the system sees as inactivity. When you play a game, that's considered inactivity, because in effect, the game file is downloaded to your computer and doesn't run on our server.
It's a cookie issue, essentially. What happens if you just leave a TCS window open for the same duration of time, without posting or viewing threads? Does it also log you out?