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.