The standard treatment for burns seems to be to put the burn in cool water for 10 to 20 minutes.
It seems sensible to me that some burns can gain a benefit from this action when the skin/flesh is still extremely hot and is essentially still cooking from the residual heat. For example, if a hand was dipped in a deep fryer, putting it immediately into cool water would be a great idea as it will immediately start to cool the the flesh. The burnt flesh for such a case may initially as high as 180 degrees C and clearly needs to be cooled as soon as possible to prevent deeper flesh getting burned.
I'd guess that the temperature at which cells in your body are damaged is probably in the range of 45 to 55 degrees. Once the temperature of flesh drops below that point (whatever it may be), perhaps there is no further benefit in cooling the burn, except perhaps for comfort.
A less severe burn caused by hot water (lets say 75 degrees C) would cool much more quickly by itself (in dry air anyway) thanks to the evaporation of the water (unlike a burn caused by hot oil), and if submerged in cool water would cool to an acceptable temperature within just a few seconds. Is there really any benefit to leaving it in cool water for 10 minutes or more?
If my understanding is correct, there is also a reduction in pain when a burn is not exposed to air. Presumably this is the reason why wrapping with plastic cling wrap (after applying cool water) is recommended by some for reducing the pain from a burn. I think this effect would also intuitively cause people to think that submerging the burn is giving benefit, when perhaps it isn't.