I've been reading about tree fertilizers and urea constantly comes up as a good source of nitrogen for trees. Let's say your dogs always pick the same tree to urinate on every day. Would that tree be healthier, grow faster etc. as a result?
In one word, no.
Dog urine contains high concentrations of nitrogen and salts. The concentrations are sufficient to burn grass roots. Tree roots near the surface are only slightly woodier and just as subject to burn.
For mature trees they can probably deal with the root burn in localized areas. A freshly planted tree might have a harder time depending on the number of dogs.
Edit: @David High nitrogen levels kill roots in a small localized zone. After the urea has been broken down into nitrogen that is available then new roots can make use of it. Dogs tend to be consistent on where they urinate so you get a cycle of root burn followed by new root growth in the kill area and then more root burn. This does not help a tree grow.