What are the rules and restrictions with using emoji or Unicode in an iOS or Android application?
Do you need to purchase any specific licencing?
As you may be aware, Emoji is a set of Unicode characters, just like A, $, 7, + and \ are. They are rendered using fonts, just like how A B C will look differently in Comic Sans, Arial and Helvetica. This is why Emoji in the same block of text looks different on Android, iOS and Facebook; they're all using different fonts for their emoji.
If you are creating a block of text (maybe in a web application, non-graphic advert or information page), you can insert the Unicode characters for Emoji and the device will do the rest. You do not need to play around with copyrights and licenses, as these are handled by the manufacturer. iOS Emoji is created by Apple, Android Emoji by Google, etc.
However, and a big however: You begin to tread into the realms of copyright if you begin to use graphics of these emojis in media such as posters, videos and anything else NOT rendered in Unicode (i.e any graphics). The actual Emoji pictures are copyrighted either as software and/or creative works. If you want to use them legally, you will need a license, or to opt for Open Source alternative such as https://www.emojione.com/ (I actually love these!).
Alternatively, have a shot at contacting Apple (or alternative) regarding obtaining a license at http://www.apple.com/legal/contact/ (or alternative).