First, I think it's important to get some agreed-upon terminology.
When I see the word emojis (which is the word you use in the title of your question), I think of characters such as these:
whereas I use emoticons to refer to text-based caricatures such as these (including the one you have in your title):
:-) ;^) :D :-( o_O (°o°)
Because your title uses the word emoji but displays what I would call an emoticon, it's hard to tell exactly what you're asking about.
That all said, I can tell you my personal thoughts* about this matter:
- The use of emojis on the site would be strongly discouraged
- Emoticons can be used, but they should be used relatively sparingly
- A good time to use an emoticon would be when it conveys meaningful information about the tone or underlying meaning of a comment (such as "that was said tongue-in-cheek")
- When compared to other SE sites, emoticons might be especially helpful on ELL because we are perpetually dealing with language barriers (sometimes it might be hard for a non-native speaker to understand something was said in a joking or friendly way)
- Be careful about using emoticons such as :-P (because some emoticons, such as that one, can be interpreted as disrespectful)
- For the most part, emoticons are less appropriate in questions and answers than they would be in comments
*Even though I am a mod on this site, these opinions are my own and do not necessarily reflect those of the mod team