I really want to solve this mystery - no scientist is going to hook up people to thermo-sensors their entire lives hoping one combusts, so we will never get the evidence stubborn science accepts, but we can research into possibilities to deduce if it really is biologically possible!
Here is an example scenario: a build up of nitrate (or a similarly reactive chemical such as ROS or hydrogen peroxide etc) in the body. When a person sweats out, the nitrate concentrates as the sweat dries on the skin. Nitrate is a very reactive oxidizer, and it causes the person's skin to catch on fire as soon as the nitrate infested sweat dries.
To quote Keith: The case where a woman's back began smoking is more interesting and may be a unique pathology, unrelated. There is a mechanism that the body uses to metabolise brown adipose fat that does not involve the thermo- regulation mechanisms of circulation of blood and body fluid. Conceivably this could runaway but there would need to be a tumour or some way that the sensory nerves were inactivated.