This article from Slate is quite thought-provoking. It reminds me of the "Doomsday Machine" from Dr. Strangelove, with one important difference: The Doomsday Machine doesn't exist while the "Letter of Last Resort" does.
The author raises a very interesting moral paradox: For a nuclear arsenal to be an effective deterrent, one's enemies need to believe that if struck first, the nuclear arsenal will be used in a devastating counter-attack against that enemy. But, once your country is destroyed, most would agree that a counter-attack against the enemy would be immoral. The paradox is that a moral actor would (presumably) refrain from using a nuclear arsenal in a counter-attack, but if one's enemy knows that, then the possession of the nuclear arsenal provides no deterrence and the enemy could stike with impunity.