authentication cookie
The loggin issues are connected with the cookies that you set. Usually a move from www.
to non www.
can make the cookie lost if domain is not set,
or a move from secure https
to non secure http
and vice versa.
Make sure that you have set the domain
(with out the www.
) on this sections on web.config
as:
<httpCookies domain="domain.com" />
<roleManager domain="domain.com" />
<forms domain="domain.com" requireSSL="true" />
session cookie
One other issue may be a connection of the session with the login, and if its lose the session is also lost the login.
In this case you must check the expire and recycle of the session. If its in memory consider to move it on database.
secure your login
Also consider this question Can some hacker steal a web browser cookie from a user and login with that name on a web site? for more settings on the cookies.
other links
Related answer Multiple applications using same login database logging each other out