Full colour emoji fonts are now available for Linux. To install:

wget https://github.com/eosrei/emojione-color-font/releases/download/v1.3/fonts-emojione-svginot_1.3-1_all.deb
sudo apt-get install ttf-bitstream-vera
sudo dpkg -i Downloads/fonts-emojione-svginot_1.3-1_all.deb

The font configuration file installed by the fonts-emojione package seems to break font substitution rules. For example, after installing the colour emoji package symbols do not display correctly in Mathematica. The only solution I found so far was the removal of the emoji font config file:

sudo rm /etc/fonts/conf.d/56-emojione-color.conf 
sudo fc-cache -f -v

Font rendering in Mathematica now works correctly and nothing else appears to have been broken. However, let me know if you know of a less radical solution.

To test the new font, visit http://eosrei.github.io/emojione-color-font/full-demo.html

Also see: