I have made some changes which have a chance of fixing this. I'm sorry I can't confirm 100%, it's hard to test if it's worked or not without putting it live and seeing if it works.
The problem is caused by the cookie domain as far as I can tell, it works fine on my dev machine but not on the live server.
If you logout, and login again checking the remember me box please let me know if it now works as intended. If remember me isn't checked, you will be logged out after 2 hours.
Please note, if you login from another device/browser you will be logged out from all other devices/browsers. I'm looking into changing this, but don't hold your breath.