Question 11: How do Shinigami Die?

Answer:
"The rule about killing to save a human] is the only one that if broken results in a shinigami's permanent disappearance.... .. It is mentioned, however, that some violations result in a more painful death than that of Rem. "

So the one and only way a Shinigami can die is if they save humans. But they can also die from other things. Nice.