I found the Emoji One font, which supports all the emojis up to the current Unicode version, including color emojis (by embedding SVGs or something).
If the font is specified, Firefox supports the colored emojis fine:
and Chrome and IE support its monochrome fallbacks:
but just installing the
EmojiOneColor-SVGinOT.ttf font into Windows is not good enough to get rid of the boxes, apparently because other fonts are preferred over it?
Installing the Emoji Keyboard (2016) extension and enabling "Auto-Replace" in the options (which is on by default):
produces colored emoji in Chrome up to Unicode 9. (The supported Unicode version is stated on the eosrei/emojione-color-font Releases page):
The auto-replace works on every site, but not in editable text boxes.
Firefox displays the characters with Arial regardless of what you have installed, which you can see by installing the FontFinder extension.
A Greasemonkey script will add the font to the body tag of all pages, but this doesn't fix the problem on several that I tested. It works on a few, like superuser.com.
Changing the browser's default font to Emoji One does seem to work in most cases, but may have side effects. (For instance, this replaces the ™ sign, too, which Chrome's extension does not do.)
The Emoji Everywhere extension kind of works, but seems to slow down page load compared to the font, and only supports Unicode 6.1.