Search for and remove cookies matching "thefiringline".
Tools menu -> Internet Options
Browsing History (middle) section, click on "Settings"
Click on "View Files"
Sort by file name by clicking on the "Name" heading in the first column.
Scroll down until you see files starting with "cookie:"
Find cookies where the second column contains "thefiringline.com" or "www.thefiringline.com" and delete all such cookies (select the right lines by ctrl+clicking in the first column, then right click, then pick delete).
Log in again, and see if you still keep getting logged out.
Imagine a pathologically idealistic, anti-religious, culturally-relativistic, iconoclastic, socially-irreverent, sociologically-progressive signature line here. Or don't.