Not a single dedicated, representative furry emoji that I'm aware of (unless we count 1F43E – paw prints), but there are a couple emoji which happen to be furry. In addition to the plethora of animal emoji (which aren't furry, but likely to be appreciated by furries), there are several emoji of cats with expressive faces (which do technically classify as furry), such as:

1F638 – grinning cat face with smiling eyes; 1F63B – smiling cat face with heart-shaped eyes; 1F63C – cat face with wry smile; 1F640 – weary cat face.

There was also the unofficial and very narrowly supported combination of ‘hacker cat’ and a few others:

https://emojipedia.org/hacker-cat

I'm not aware of any single furry emoji, but I've noticed a lot of furry communities on the Fediverse using custom (non-Unicode) emoji packs; see [1] for how that might work. Also an example of the feature in use[2].

[1]: https://misskey-hub.net/en/docs/for-users/features/custom-em...

[2]: https://thegayagenda.fans/notes/ah5imm41r1ru05a7