Mozilla's official Firefox Flatpaks distributed through Flathub and Flathub-beta are unable to copy text to the system clipboard on Wayland.
Copying text from the URL bar or in HTML input fields, using Ctrl-C or the copy option in the right-click menu, seems to update the clipboard only for Firefox; pasting the text into the Firefox URL bar or into other HTML input fields works fine.
However, outside firefox, the Wayland clipboard is set to an empty string even if it was populated by something else originally. In other words, copying text in Firefox clears the system clipboard.
Pasting into Firefox from the system clipboard works just fine.
Interestingly, the JS clipboard API does not have this issue; a webpage that updates the clipboard using JS correctly updates the system clipboard for pasting into other applications.
This bug was originally reported two months ago for Firefox 88.0 on Reddit, and I just encountered it after trying out the Firefox Flatpak: https://redd.it/myften
No other Flatpak app I have tried seems to have this issue. I've tried Epiphany, Chromium, and even Fedora's Firefox Flatpak package using the Fedora runtime; this issue seems to be specific to Mozilla's Firefox package.
I've replicated this bug on Firefox 90.0.2 from flathub and on 91.0b6 on flathub-beta.