I've been able to install FBConnect successfully, and it's working fine on all browsers I've tested except Firefox 3.0.11 (on WinXP).
When I log in via the Facebook Connect button, all seems to be going fine. The page reloads; the 'Logged in as' box appears in the corner of the screen; my Facebook profile pic comes in... then, for some reason, the page immediately reloads again. And we're back to where we started.
The only clue seems to be a couple of JS errors reported by Firefox:
aNode is null
chrome://browser/content/nsContextMenu.js Line 309
this.editor is null
chrome://global/content/bindings/textbox.xml Line 152
I've tried it in Firefox's safe mode, and the problem still arises.
But it works perfectly when I try it on IE, Chrome or Safari on the same machine; I've also tested on Firefox 3.5 (on another machine), and that was fine too.