That's not good. Could be a problem with cookies or it could be a problem with the system being able to write to the server.
Which browser are you using?
To see if the session is being saved on the server, take a look in your data folder (I believe it's /data/_sessions/ ). When you log in, your session should be saved and the timestamp of the appropriate file will be updated. If the timestamps aren't changin, then the system isn't able to write to the server.
Just to be clear, after you delete the files in the _session folder, then try to log in again, there are new files in the _session folder?
To be honest, I'm kind of stumped, but there a couple of things we can look at/try:
Sorry to not have a more solid response... I would really like to get it fixed for you though!
I've put together a file that can help us narrow down the problem by outputting some relevant messages. If you don't mind uploading it to your server then reporting back with the results, we should be able to find the exact location of the problem. The file is a modified version of the common.php file which is located in the /include folder.
I'd be curious to see the messages before you log in, just after logging in, and then from the following page where you are no longer logged in. They will all look something like this:
Really sorry for the trouble.