I have a calf hair rest and side plate on my bow and it appears that some of the hair is wearing off. This has not seemed to effect tuning as it is a very thin layer of hair over the hide. My question is can I just shoot with the hide left on my bow or is the hide to thin and will wear out quickly?

Might be a brace height issue. How long has the hair rest be on ?

You can shoot off a leather shelf pad.  The hair just makes it smoother.  One thing that eats up shelf hair quickly is the leading tip of the fletch.  I smooth that with a razor and "ramp" a bit of Duco on the quill for a smooth contact point.

I don't think it is a brace height issue as I get perfect flight bareshafting even the way it is now with just the leather on it. My question is will the leather work for a rest or do I need to replace it?

The leather will work fine. You don't need to replace it.

Don't fix anything that isn't broke.