Seems like a Cookie issue... what security software do you have installed, if any?
Some Security Suites, like BitDefender (for example), monitor Cookies [for Tracking cookies, etc..]. If a particular setting is adjusted within BitDefender, it will disallow all cookies.
ALSO, do you have your FireFox setup to "Clear History on FireFox Exit"? If so, that too might be an issue. Have a look at THIS
. Make sure that the "Cookies" box is unchecked. Doing so will allow you to keep your log in sessions logged in, until the Cookies expire.
If the forum boards you're visiting are SMF or vBulletin, FireFox handles the login details for those particular forums strangely. If you have allowed FireFox to "Remember Username and Password" for the particular forum you're visiting, simply double click in the Username field on the forum, then select your username from the dropdown list. It will automatically fill in the Username & Password fields.