I just tried downloading the latest Firefox Nightly from https://archive.mozilla.org (http://archive.mozilla.org/pub/firefox/nightly/2022/11/2022-11-05-21-53-50-mozilla-central/firefox-108.0a1.en-US.mac.dmg), installing it to /Applications, then running it (by double-clicking on it). I had no problems. This was on a fresh install of macOS 13 (build 22A380, the current release), on an Intel Mac. I'm sure I've done something similar on an Apple Silicon Mac, but I haven't yet re-tried it for this bug. That will follow shortly.
Bug 1799332 Comment 2 Edit History
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I just tried downloading the latest Firefox Nightly from https://archive.mozilla.org (https://archive.mozilla.org/pub/firefox/nightly/2022/11/2022-11-05-21-53-50-mozilla-central/firefox-108.0a1.en-US.mac.dmg), installing it to /Applications, then running it (by double-clicking on it). I had no problems. This was on a fresh install of macOS 13 (build 22A380, the current release), on an Intel Mac. I'm sure I've done something similar on an Apple Silicon Mac, but I haven't yet re-tried it for this bug. That will follow shortly.