Hi Kristing,
I think a lot of dog peeing will take it's toll on most tree's and shrub's although some may be more resilliant than others, for instance I've noticed my dog tends to c0ck his leg up on my lavender bush on a daily basis, with no apparent ill affects.
Another way to combat this is to dig out the dogwood and plant something in a large tub or pot so that the dog pee hits the pot first and runs off, I think a blueberry bush may be well suited to this if planted in (acidic) ericasious compost. Good Luck. Tbird+