I just died from this while trying to cook my rabbit and cow steaks :(
I was healthy, hydrated and energized, I ran to a pond a few metres away and rolled around in the water, but still came very close to death. Then I went back to the fireplace and it happened again instantly even though I was drenched, and this time I died in less than two seconds.
The status messages ("my face is melting" etc.) are the only indication that you are on fire, so if you're not paying attention (e.g. because you're monitoring cooking meat) you could easily die without even knowing what happened. One might expect some screams and flames.
This was a fireplace in a stove in a house, if it makes any difference.