So it is possible to actually get burns by accident?
I think this is the rarest type of injury, even less frequent than frostbite (which requires no clothing on a body part and a really low temperature).
As for dogs, I'm not sure they would be burned at all. I frequently call forest maidens, which requires a burning stack. Since my char is always surrounded by animals, and they cannot move aside (because of crowding), they often land on the burning tile when my char goes to sleep. They never get damaged. Maybe this is handled separately by game code.
Anyway, I found out that burning injuries are the most convenient ones for training Physician skill. This is because they give opportunity to either apply a compress (minor wounds) or both clean and apply a compress (serious wounds). Applying a compress has a +2 bonus to learning. Cleaning doesn't have such bonus, the same with treating frostbite. Another one treatment which has such high chance of learning is stopping bleeding, but bleeding injury is both difficult to obtain and dangerous.
Using cheap armour allows to protect selected body parts (except eyes, unfortunately) against too much burn damage. It works for every type of damage, but is easier for burns.
Previously I used bruises from falling off trees, but they apparently always have only the compress treatment available, never cleaning (because they are blunt damage) and even though cleaning is just a standard chance to learn, it's still better to have two chances per injury per day than one.
The only inconvenience is that all burn damages, including minor, are yellow level status at maximum when treated by player (even with all-healing herbs), and can be upgraded only to celadon level by sage, the same as frostbite. Which means than it won't heal so quickly if unlucky. But that could be the point, if used for learning.