I've seen wood being burned for spears in movies. I've also seen people burn wood for Sho Sugi Ban finishing technique. But....
Does burning wood increase any mechanical proprieties?
Are there any other advantages to burning wood?
It's not the burning of wood, it's the heating of it.
By heating it one of the things that happens is you remove the moisture from the wood. This makes it more physically stable. Letting a stick dry out in the sun will have similar properties (as long as it isn't in the rain).
Moisture helps keep wood supple and when you want to push a stick into someone/thing that is not a good quality to have. So heating removes moisture and sets the cellular structure.
Not burning or charring the wood, but heating it to a high temperature.
In this article dealing with the "Strength properties of thermally modified softwoods..." that deals specifically with three different softwoods the authors present information about distinct changes that occur in the mechanical properties brought about by heat-treating.
First, the authors discuss the reasons for heat-treating (a well accepted and common two step process that takes the temperature up to the range of 200 C. - think baking a pizza):
Then they discuss the specimens, testing methods and mechanical test results that reveal these changes (plus others):
Finally, there is discussion about the likely causes of these changes which are caused by activity at the cellular and molecular level.
Surely, we should not extrapolate these particular results to apply to all woods (no hardwood testing here) but I think it's safe to say that the extreme heating of wood does permanently change mechanical characteristics of wood. (Increase or decrease? - some one, some the other.)