The subject pretty much says it all. After upgrading my iPhone and iPad to IOS 14, I changed the default browser from Safari to Firefox. When I open an entry in LastPass and click Launch, I get a message – “Username copied to clipboard” but then nothing happens. If I change the default browser back to Safari, Launch works fine. It also works if I change the default browser to Chrome
Is this a LastPass problem or a Firefox problem?
Firefox for iOS doesn't support extensions, which means you would have to switch between Firefox and the LastPass app if you want to insert your passwords into Firefox for iOS, or use the copy and paste feature as you've seen to paste passwords in to Firefox from LastPass.
The Chrome or Safari browsers will work, as you've experienced, with the Launch function.
It is not relevant if the browser supports extensions. All mentioned browsers use the Blink engine because Apple makes that mandatory. So this does not explain one browser working better than another.
LastPass < 4.8.8 had a lot of issues on IOS. Hope it gets better again now.