I have never heard of any instance where a person died simply from being kept from sleeping, e.g. in a POW or interrogation situation. However, sleep is definitely required to maintain mental alertness, so in a survival situation, as you mentioned, there's a much higher chance of an accident or making a dumb decision if you're severely sleep deprived.

I remember watching a TV show about a sleep deprivation experiment. The test subjects were able to function physically just fine for a long time. It was the mental functioning that rapidly deteriorated, and which seemed to require sleep. However, note that there are people who do not sleep at all. I'm not sure how their brains cope with the lack of sleep, though.

I'm not sure that sleep really fits into the rule of 3's analogy. Strictly speaking, it's absence is not deadly the same way that lack of oxygen or water is deadly.