Tadas,
The auth_token will expire from Facebook after 24+ hours, normally. JFBConnect should refresh the token anytime the user logs in through Facebook on your site (clicking the button, auto-login, etc). If the user isn't logging in through Facebook, just using the Joomla login, then the token will likely expire.
There is a way to get an extended token for a user, but JFBConnect doesn't currently support that. Those tokens will last "a long time" (likely 1 month plus, but Facebook is intentionally vague about the length).
Finally, all of the above is assuming you're using the PHP methods to push the Open Graph actions. If you're using Javascript, there's other things to consider that we can help with.
Let me know if anything above rings a bell, or if you can explain any more about how the user is logging in in the first place or how long "after a while" is for the token to expire.
Thanks!
Alex