I'm currently running Elementary OS 5.1.7 Hera, and just downloaded Firefox Developer edition, which runs independent to the regular Firefox, so Live Server extension doesn't recognize FDE.
The Live Server extention opens the localhost tab on your default browser, but you can copy the URL into any other browser and it will work, as long as the server is running.
If you want the tab to always open in Firefox Developer Edition, you can change the settings of Live Server.
In VScode, go to File > Preferences > Settings (or Code > Preferences > Settings on Mac)
In the search bar, type liveServer.settings.AdvanceCustomBrowserCmdLine
Click on Edit in settings.json
change the value of "liveServer.settings.AdvanceCustomBrowserCmdLine" to the location of Firefox Developer Edition (for example, C:\Program Files\Firefox Developer Edition\firefox.exe)
Note: Live Server also has a CustomBrowser setting, which is simpler to edit, but it has limited options and Firefox Developer Edition is not one of them. AdvanceCustomBrowserCmdLine overrides CustomBrowser.
You can read the Live Server docs here.