I mean you could, but how would that work? Those "emojis" would end up only working with that particular font, so you'd select a FA icon, paste it somewhere, and you end up with some random character that has nothing to do with the icon.
The point of real emojis is that they are standardized and wildly adopted unicode codepoints that should in theory represent the same thing no matter what font is being used. You can paste a π€·ββοΈ and it doesn't matter if you use the Noto fonts, or Microsoft's fonts, or Apple's fonts, or Samsung's fonts, you'll always end up with a guy shrugging. Icon fonts follow no standard, so a gear icon in FA might end up being a bell icon or something else unrelated.
What you might be after is a font viewer that lets you see all the glyphs of a given font that you can then copy paste the codepoint where appropriate. But even then usually icon fonts are used with a matching CSS stylesheet so you can add them by name rather than by character, and it fixes some accessibility problems with screen readers along the way by just not having a random H that's supposed to be an icon.