What does yeast do to make root beer?

No.

Root beer was at one time made in essentially the same way as beer (MASH up a bunch of stuff, let it ferment), but it has never been made out of beer.

Modern root beer is made like all other sodas: by adding flavoring and sweeteners to carbonated water. Under the old method, root beer could be (very mildly) alcoholic; under the new method, it's completely non-alcoholic.