This is not going to be a one shot solution. Lets make the diagnosis right first. I am confident that lawn is not getting burnt because of dog
pee (or excessive nitrogen), but there is lack of some factors to balance the impact. Those factors might be those that make a lawn healthy (things that can make unhealthy such as: lack of soil creatures -worms, bacteria, fungus or many many others, mono culture of 1-5/6 different grass types, low levels of soil
carbon ratio, low levels of unbroken organic matter, lack of drainage (for clay) or very limited watering), or you might have too many animals in a small area. So, try to make your lawn healthy and as diverse as possible. Leave you lawn clippings on the spot, dry grass will buffer the impact and will create a mulch layer. Higher levels of soil carbon ratio helps a lot but you will also need unbroken organic matter (think about mob grazing). Let weeds grow! Every plant has different needs and also tolerances. It won't happen just in one year, but it will get better and better. Like 4 years, I guess -didn't take notes, after I implemented the recommendations given on this site (and rich soil also), classic lawns problems were gone. No need to mow every other week, no need to water (maybe once in a month for our Mediterranean summers), no burns, no compaction, and not many weeds (cut them high).
In meantime, you can water the spot well (as you do when you burn by your lawn with excessive nitrogen by chemical fertilizers), throw some seeds and drop some smaller sized
wood chips (or some shredded brown leaves) or
compost.
I see you responded to a similar
thread, how that worked out for you?
Brown pee spots