There comes a countdown moodlet of some sort from fire traps, but that is similar to the moodlet you normally get (after which you get all black and burned). Mummies DON'T die when that moodlet has counted down, they don't even become burned (they may still faint as normal sims).
However, from a real fire they get a very short countdown moodlet at like 20 sim minutes. When that counts down, they die.
I have not tested this aspect exactly, but I have reason to believe that they can run away and put themselves out before it counts down, if a dive well is near. These countdowns are not the same and if we are to talk about when they actually die, we need to distinctly differ these (the moodlet pictures of them aren't even the same).