The biggest problem I see here is technical implementation. Even ignoring the fact that some questions could be specifically discussing emoji, it'd still be difficult to implement.
Take a look at your phone keyboard's emoji section. If you're on Android, it's a ton. If you're on iOS, it's practically infinite. And it's still growing. There are new emoji added every year.
It's not possible to just blacklist "all emoji." Emoji are part of the Unicode standard, so you'd have to also block all of Unicode for an umbrella blacklist to work. The other option is to manually maintain a list of emoji in the form of disallowed Unicode characters. Not only is that tedious, there are instances where emoji are actually combinations of different emoji or even other Unicode characters.
Now sure, it's possible to build and maintain an emoji blacklist, but Stack Exchange is having enough trouble with making the homepage responsive, thanks to limited resources.
How many times have you seen actual emoji in a post or comment where they aren't relevant? I know I haven't seen it very often at all. I'm pretty sure I've seen emoticons much more often (
:) and such).
Considering the technical challenge, combined with the low frequency of it actually happening, I don't really think the benefit outweighs the cost here.
This is all ignoring the possibility of legitimate uses of emoji, the fact that some people are actually fine with them in posts, etc.