This only seems to happen with the Save File dialog, not the Open File dialog.

Not able to replicate the issue. see screenshot Steps to replicate the issue please. Does it happen when downloading anything? What security software are you running? Did you try with Firefox in Safe Mode? https://support.mozilla.org/en-US/kb/diagnose-firefox-issues-using-troubleshoot-mode#w_how-to-start-firefox-in-4troubleshoot-modesf5safe-modesf

Do you happen to use the Windows On-Screen Keyboard (OSK)?

Do you have any software that adds shell extensions to the File Manager context menu?

Try clearing the Most Recently Used (MRU) lists with a file cleaner or MRU-Blaster.

It appears to only be triggered when trying to save an image via "Save image as...", or when the Free Download Manager extension falls back to using Firefox to download a file.

I only use Windows Defender.

This issue happens in Safe Mode, as well (excluding the Free Download Manager issue).

I don't use the On-Screen Keyboard,

If you're talking about right-click items in Windows Explorer, then yes, but I don't see how that would be the issue.

Shell extensions get loaded into Firefox when opening the file picker which can cause issues.

Did you try using MRU-Blaster or CCleaner to clear the MRU list?

Does it still happen if you reboot in Windows Safe Mode?