But urine is very high in nitrogen and could potentially "burn" the plant. What are some common plants/trees that can flourish with urine as a fertilizer? Is there some way to tell which ones to avoid, lest they shrivel up and choke on the nitrogen?
Urine is very high in nitrogen. One can observe how plants handle the quantities humans would produce by observing the yards of people with large dogs. Plants usually get "burned" from the quantity of nitrogen deposited by the dogs.
If your compost heap has a lot of straw/cellulose, then it probably needs more nitrogen. If you want to use your own stuff to fertilize plants, mix it about 50% with water in a bucket (or watering can) then apply it to plants. Fill the bucket/watering can indoors, and wash it out afterwards.