Ah, in my case it's definitely a Firefox issue.

When I brought up the TE site in Internet Explorer, logged in and clicked the "remember me" box, and then shut down IE, when I re-open it and access the TE site again I'm still logged in.

I've been trying Oakleaf's suggestions (thanks!) and playing around with my Firefox privacy / cookie settings, but so far no dice -- whenever I close Firefox & then open it up again, I'm logged out of TE.

Eh, I'll figure it out eventually.