I tried all the usual things: clearing the cache and cookies, temporarily disabling extensions, and so on. Nothing made a difference. Some other people on Twitter have experienced similar problems, and one of them suggested using Tor, which I have but hadn’t tried because it’s simply Firefox in plate armor. I tried it, and the problems didn’t occur there! (Though since Tor uses proxies, Twitter started sending me email warning that I’d logged in from an unfamiliar IP address.)
Then I tried using Firefox from a different account. (I have one account with administrative privileges, and another from which I do normal work.) That one didn’t show the problems either. This got me thinking about root certificate issues, but Firefox on the Mac stores its root certificate with the application, not the user information, so that wasn’t a variable. But then I thought about my profile and decided to create a new profile. Running Firefox with the new profile, I didn’t have the problem.
This was a slight nuisance. I had to re-add my plugins, import my bookmarks, and “void my warranty” (a dumb joke that some managers take seriously) by changing about:config settings to my liking. Of course, I did this one step at a time and made sure I hadn’t broken anything. Now things are the way I like them again and nothing broke.
I still don’t know what in my profile was causing trouble, but I’m posting this as a possible fix to some Firefox problems.