We are still trying to verify this issue and we need more information:
1. When you login facebook do you checked the "Automatically Login" box
2. When logout happens, what's the link you opened? Some profile page of your own, or you are browsing your friend's photo etc? Sometime when you try to open a profile related page, facebook may consider it as a high-security page, in this case, it may clear all your login info and ask you to login to verify it's YOU trying to open your own profile page.
3. In our testing, IE has the same issue. Facebook logged out when we open the 4th tab after login. Could you please try IE, and let us know if there is the same issue. If we verified IE has the same issue, we need to try a different approach to resolve this issue.