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Rem is one of the few Death gods to discover how to kill a Shinigami. If a Shinigami uses the Death Note to extend the life of a human they care about, they will die since they have failed in their eternal mission of shortening human lives. This was stated by Rem when she claims she would kill L for Light; she initially states that it would make no difference as it was to protect Light, whom she did not particularly care for. However, since she then had such an attachment to Misa (whom she was convinced would inevitably die if L was allowed to live and suspect her as Kira), she would die. Rem reveals this to Misa, who later reveals it to Light and Ryuk.
A Shinigami could end up killing itself as well, if it neglects to continue filling its Note with the names of humans. If it fails to do this then it will cease to lengthen its lifespan, and eventually run out completely.
A Shinigami may be killed by execution as well, in the Shinigami Realm. If a Shinigami breaks the laws of the Shinigami Realm, it is subject to one of nine levels of punishment, starting with level eight at the lowest infraction, going up to level one, and then the "Extreme" level. If a Shinigami is subjected to levels three or above, it will die after it is finished being punished. This is why Shidoh stops himself from telling Mello where the other Death Note is, after reading this rule in the scrolls he was given for his mission to retrieve his own Death Note, as he would be punished with "second degree" punishment and die.
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