Sorry to bump this thread, but something that worked for me was creating a "fonts" subfolder in Thunderbirds' own folder and placing EmojiOneMozilla.ttf in that folder.
First, download the font itself.
Second, create a "fonts" folder (you need root access for that):
sudo mkdir /usr/lib/thunderbird/fonts
Finally, place the font file in that folder (mine was saved in ~/Downloads, yours may be in some other place, so change the next line, if needed):
sudo cp ~/Downloads/EmojiOneMozilla.ttf /usr/lib/thunderbird/fonts/
Restart Thunderbird and it's done. Hope it helps!