There isn't a name for the fear of the Koala unfortunately, even though it is quite a common fear...
Some people tend to associate it with a fear of bears, but actually the Koala is not a bear at all, it is an arboreal herbivorous marsupial.
Native to Australia, Koalas usually live for around 13-18 years and the closest thing they resemble is a wombat.
They look cuddly and cute (unless you have a fear of them of course!) but their fur is like that of a sheep and they are wild animals so do not make good pets.
I usually put a photo on my posts relating to the topic, but I hesitate to do that if you are frightened of them, but hopefully this one will be okay, it really is cute!
I'll also include a video proving just how fierce koalas can be. Even if it doesn't help cure your fear, it might make you feel a little more justified in feeling nervous around a creature that so many people only see the "cute side" of: