Both Gmail and Facebook may monitor for changes and dynamically update the page. Typically these would be small amounts of data, but some add-ons may spring into action, for example, to pre-load or download videos.

Can you replicate the problem in Firefox Safe Mode? That's a standard diagnostic tool to bypass interference by extensions (and some custom settings). More info: Diagnose Firefox issues using Troubleshoot Mode.

You can restart Firefox in Safe Mode using

Help > Restart with Add-ons Disabled

In the dialog, click "Start in Safe Mode" (not Reset)

Any difference?

Firefox has a few buttons to clean up memory usage that you might try and see whether any of them help. You could try one at a time and then check the Task Manager, or try one every hour. Hopefully this will help narrow down the nature of the problem.

To access those buttons, open a new tab and type or paste about:memory in the address bar and press Enter. The "Free memory" buttons are typically in the third box over.

But - once the dial-up connection is broken, there is no way to update anything. Why even have the browser running when the connection is hung up?

It is a good point, there is almost no reason to have Gmail open after you are disconnected.

• Firefox uses too much memory or CPU resources - How to fix

leaving Firefox running and active tabs open means that refreshing the tabs when I reconnect takes much less time than loading from scratch.

I've tried the free memory option. We'll see how it goes from here. I'm still baffled as to why Firefox eats up memory when idle and working offline is checked. BTW, I vacuumed dust out of the intake and CPU cooling fins just to make sure that wasn't affecting things too.