Start Firefox in Safe Mode {web Link} by holding down the <Shift>
(Mac=Options) key, and then starting Firefox. A small dialog should appear. Click Start In Safe Mode (not Refresh).

Is the problem still there?

While still in Safe Mode,

In the address bar, type about:crashes<enter>. Note: If any reports do not have BP in front of the numbers/letters, click it to submit them.

The crash report is several pages of data. We need the report numbers to see the whole report.

Using your mouse, mark the most resent 7 - 10 crash reports, and copy them. Now go to the reply box below and paste them in.

(Please don't take a screenshot of your crashes, just copy and paste the IDs. The below image is just an example of what your Firefox screen should look like.)

https://support.mozilla.org/en-US/kb/websites-look-wrong-or-appear-differently

https://support.mozilla.org/en-US/kb/firefox-slow-how-make-it-faster

https://support.mozilla.org/en-US/kb/firefox-uses-too-many-cpu-resources-how-fix

https://support.mozilla.org/en-US/kb/firefox-hangs-or-not-responding

https://support.mozilla.org/kb/Firefox+is+already+running+but+is+not+responding

Thank you for your email. Unfortunately I have been out of town, so I was not able to follow up until tonight.

I put Firefox into Safe Mode as you directed, and I determined that the problem exists in Safe Mode too. When available processor memory was almost completely consumed, I opened a new tab and entered about:crashes in the address bar. Only one entry was showing, clearly because Firefox did not crash. That's always the scenario. Nothing actually crashes, but so little memory is available that nothing can run, and it doesn't matter if I close all my tabs. I can move the mouse from one application to another, but no application comes to the fore. Restarting the system is the only option.

Mike.At.Home said

(I just found out today (Oct. 29, 2016) that Firefox will not be maintained on Mac OS-X 10.7.5, but I still wish to stay with 10.7.5 as long as possible.)

For the Release yes as Firefox 49.0 and later requires OSX 10.9 and later now. There is a option is getting security and allowed stability fixes for a while still for OSX 10.6, 10.7, 10.8 users.

You can use the Firefox 45 ESR (Extended Support Release) as ESR channel versions generally gets eight updates so 45.8.0esr will be out when Firefox 53.0 is released. https://support.mozilla.org/en-US/kb/firefox-osx

How to crash Firefox; Instructions here; https://developer.mozilla.org/en-US/docs/Mozilla/How_to_report_a_hung_Firefox