If by my death I could save another's life (like falling on a grenade) do I have a moral obligation to do so? Are there circumstances when this might or might not be true? Are their schools of philosophy or specific works that address this question?

One can perhaps imagine circumstances in which it made ethical sense that someone should sacrifice their own life for that of another person. For instance, one might sacrifice oneself to save one's child. We might feel such an act is an especially good act, a 'supererogatory' act i.e. something above the call of duty. But the idea that one might have an OBLIGATION to sacrifice one's own life for another is a much stronger idea. I doubt there is anything much in Kant's moral philosophy - a moral theory that organizes all acts of moral worth as done out of a motive of duty - to entail the duty to sacrifice one's own life for that of another, but a moral theory that surely does speak to this idea is consequentialism. Basic forms of consequentialism say that the rightness or wrongness of an action consists exclusively in the goodness or badness of its consequences (so motive is irrelevant). Now this tends to generate a standing obligation to maximize the good (however conceived - human happiness, preference satisfaction...), and that standing obligation would entail a moral obligation to sacrifice one's own life WHENEVER doing so would cause less human pain/unhappiness/preference frustration etc. than not doing so.

Your question thereby throws up two of the cardinal problems with consequentialist moral theory:

1) it is too demanding (the standing obligation to maximize the general happiness/preference-satisfaction etc) is too burdonesome and doesn't leave enough room in life for other values and projects

2) we're not generally in a position to judge which of two prospective actions that present themselves (throwing oneself on the grenade or not) is the better maximizing strategy.

Both of these criticisms inspire 'rule' or 'indirect' consequentialism: the form of consequentialism that says our standing moral obligation is not to maximize the good with every action, but rather to adhere to a set of rules or practices which, taken overall, constitute the best collective maximizing strategy.

Read another response by Miranda Fricker
Read another response about Death, Ethics