Interesting idea, but probably harder than it seems like it would be.
There are many people on here who have chronic fatigue syndrome and other health problems that cause them to have extreme exhaustion all the time (myself included) And yet they don't die.
On the other hand, the kind of exhaustion you're talking about seems to be to just run around until you collapse. I'm not sure how easy that one would be to do either because I had an eating disorder for a while called
Exercise bulimia where instead of making myself throw up as a regular bulimic does, I would exercise until I was so exhausted that I would have to crawl from the room where my elliptical trainer was to the bathroom at the end of the day. I would exercise from 8 to 10 hours every single day no matter what. Then I would crawl and lay on the floor in front of the bathroom for hours just trying to find the strength to get up and take a shower so that I could go to bed. Though I wished I was dead during those times, I never died. Perhaps if I had kept going until I just collapsed completely, I might have died.
It seems like it might be one of those things that seems like it's really simple, but actually turns out to be much harder to do in reality. I'm assuming your body would have some kind of defense mechanism where you would collapse and then wouldn't be able to get up so that your body could rest?
This is just my best guess. I'm certainly not a medical professional.