Felling trees is a dangerous task to begin with, but that task is even more dangerous when the tree is on fire. In both cases there's a danger of limbs falling while you are cutting, the tree not falling the way you intended either because of hidden decay or other structural problems with the tree, and the phenomenon of 'barber chair', where the tree splits and kicks back at the sawyer, which can be fatal.
Why do firefighters cut down trees that are burning? Usually it is because the tree threatens to breach the fireline, or because it is close to where fire crews are working. The challenge of cutting down a burning tree is of course standing that close to flames, but there is also the risk of the gas in the chain saw catching on fire. Watch this video to get a feel for the danger and risks involved.