Whether the emoji is colour or not, depends on the combo of Browser+OS+Font. On Linux, pretty much every mainstream browser renders emojis in colour. On Windows, it's a bit more complicated: Windows 7 always renders emojis in black and white regardless of the browser, whereas Windows 8.1 and Windows 10 render them in colour. Unless it's Internet Explorer, where they are in black and white again.
A LOT of sites also do 'on-the-fly' unicode emoji swap-outs for little png images. Twitter and Facebook are a good examples of this. That way the emoji is always in colour regardless of the platform.