Oddly enough I have never had a Dog pee on any of my mounts but my Wife has a Cat that urinated on a Coyote hides that had fallen off the wall. I didn't know it until a few days later. I could smell it. So I tried washing that area with a mild soap and water. Well I ended up throwing it away a few years later because that spot ended up becoming fragile and tore really easy. My old Taxidermist, who has since retired, said that it was a result of something in the urine.
I would take it to a Taxidermist and have them take a look at it and clean it if the damage hasn't already been done.