Are you trying the keyboard shortcut Ctrl+V (or on Mac, Command+V)? There is a bug with the right-click context menu in some situations, so you may need to use the keyboard shortcut.
Websites differ in how they handle pasting. Some are passive and let Firefox manage it while others intercept the paste and do fancy things with it. For that second type:
Some privacy guides recommend changing dom.event.clipboardevents.enabled to false in about:config (or a user.js file), but that can cause problems on sites that intercept pastes by blinding them to your actions. Does this sound like something that might be a factor on your Firefox?
