Firefox 78 was the last version for your OS (Firefox Mac OS X 10.9, 10.10 and 10.11 users move to Extended Support Release).

Firefox has moved on, of course, and is up to version 133, but I don't recommend trying to install anything newer than you have.

Is a website telling you the older Firefox is no longer compatible? Where are you having a problem?

Firefox 79.0 to 115 ESR requires macOS 10.12, 10.13, 10.14 at minimum to run while Firefox 116.0 to current requires macOS 10.15 or later.

If you have any questions related to Firefox on Mac (macOS) in the future, then you should post in the Firefox section https://support.mozilla.org/en-US/questions/new/firefox using this form https://support.mozilla.org/en-US/questions/new/firefox/form

I would move this thread from Firefox for iOS (for iPhone/iPad) to the desktop Firefox section as you have macOS but an Agent already processed this so I cannot move this thread now due to recent changes.

I am running a Mac Pro that I will not be updating the OS as no newer versions are available for this age of the computer. It is a 2010 Mac Pro with an OS of 10.13.6 (High Sierra). It can't go any further. Can I still run the current Firefox that is currently on the system?

For the eol macOS 10.12, 10.13, 10.14 the most current Firefox you can run is Firefox 115 ESR which is currently at Fx 115.21.0esr (released Mar 4) and will be updated till Fx 115.27.0esr in August unless Mozilla extends updates for a third time.

https://support.mozilla.org/en-US/kb/firefox-users-macos-1012-1013-1014-moving-to-extended-support

Firefox 115.21.0esr.dmg https://archive.mozilla.org/pub/firefox/releases/115.21.0esr/mac/en-US/ other language folders https://archive.mozilla.org/pub/firefox/releases/115.21.0esr/mac/