What you are describing aren’t diseases. Those are actually ‘wounds’, like a bite-wound for example, which needs to be treated by cleaning the wound for example. Diseases are stuff like the Flu.
I would like the Idea. And it would make sense too - burned tissue needs to be treated well, or one could potentially die from it - or at least the wound would get pretty bad.
On could also consider having something similar to the broken bone mechanic - A burned hand/arm won’t be able to do much of anything due to the sheer amount of pain it brings to move it, let alone grab or wield something. A burned leg wouldn’t be of much use either.
Thing is, if we get into burnt-mechanics, we would potentially have to include scar-tissue as well. If the burns are heavy enough, permanent scar tissue would remain which would hinder movement, could hurt or even open up again, if enough pressure/stress is applied.