I created an account on your server and can confirm that it succesfully kept me logged in after closing down my browser and restarting. (You may delete the account called 'cheeky' if you like)
and a few hours later, i can confirm that i'm logged out :S Sounds odd, this one. Remember that Drupal does not use any cookies of its own and just repies on the inbuilt PHP session handling cookie. It seems that drupal and the browser are both behaving correctly. Is something happening in your Apache/PHP to cause it to lose PHP sessions? Server re-configs / reloads / restarts? Cleaning of temp data?