Those are just Unicode characters so any textboxes that support Unicode can display them without problem, provided the font and glyphs are available on the system. However the look and feel for each Unicode character depends on the renderer and font used for them.
Traditionally characters are filled only with a single color. Colored emojis are quite a new thing, hence their support varies from platform to platform. Various techniques have been invented for coloring like embedded PNGs, SVGs or layered masks. But all of them require a new renderer which was not available in older OSes.
As a result since version 50.0 Firefox has embedded its own renderer and font for supporting colored emoji. You can find it in the release notes
Changed
Added a built-in Emoji set for operating systems without native Emoji fonts (Windows 8.0 and lower and Linux)
The font file can be found in %FirefoxInstallDir%\fonts\EmojiOneMozilla.ttf
Update:
Firefox later switched to Twemoji