WordPress has supported Emoji since version 4.2. The following function will disable Emoji for those that sensibly want nothing to do with them. Interestingly, Emoji support was released as a "trojan horse" to address a critical (database) security vulnerability. If interested in the back-story, check out this video from WordPress developer Andrew Nacin.
Copy and paste the WordPress function into your theme's
functions.php file or, if you sensibly have one installed, your custom functions plugin.