So I am going through the file to see what code is cauing the problem. But in the meantime I would like to know what code is essentail to have in the file.
Issues with logging in/out are most commonly caused by incorrect server configurations. In specific related to the session save path (either no set at all, or done incorrectly in the php configuration, OR the specified folder is not writable by the user). The best way to resolve this is by contact your webhosting provider.
I must say that I do have experienced some automatic logout actions, so there may be a bug somewhere. Still on my list to test.
Not sure if related, but I spotted a tiny anomily in e107.htaccess.
Change the last " to a plain one on line 86. Really hard to spot but it has the curly " (like those word auto converts quotes to).
It has this ” not the correct one "
ExpiresByType image/x-icon "access plus 1 year”
Submitted a correcting pull request - will be evaluated by devs soon.