I've been having this problem myself...pretty much forever. It happens on multiple computers and multiple installations of WordPress with different versions of the Firefox browser. Can anybody explain?
I think cody has explained it pretty well, but just to reiterate:
Firefox _will_ let me into the administrative back-end, and once there, everything works fine. It's only when on the public side of your site that WordPress seems to have no idea who you are when using Firefox. So for example, rather than presenting "Log Out" and "Site Admin" options over in the sidebar functions, in Firefox it presents "Log In" and "Register" links. Clicking the "Register" link does, in fact, take you to the registration page, yet clicking the "Log In" page brings you to the administrative dashboard.
Similarly, in Firefox the "Edit" links on the public post display will not appear, and even the Wordpress Stats plugin counts your visit as a non-logged-in user. Also, the commenting form doesn't identify you as a logged in user. And yet, I assure you, I am logged in. I can go directly to the wp-admin directory and WordPress knows who I am.
This is really bothersome, and I've lived with it for a long time now hoping that it would eventually be resolved, and I'm surprised that nobody else is talking about this on these support forums. Could it really just be me and Cody?