I *think* we fixed the problem by going in as admin and changing the global chat settings for Inactivity Timeout from default/blank (defaults to 120 seconds) to 7200 seconds. But it could be just luck, we've only had a couple hours to test the change.
Clearly a bug; seems the AJAX callbacks for polling for new messages is not properly resetting the inactivity timer.
On the other hand, if I log out of Community Server and log in as a different member, the previous account eventually times out in a couple minutes even though I changed the timeout to 7200 seconds (this is good).
So it seems that, at best, the bug is that the default is WAAAAY too short. 120 seconds is acceptable for AJAX callback timeouts. But not for chat messages; I don't know anyone on the planet who hangs out in a chat room and doesn't stop to listen for more than a couple minutes.