What is the extent of the moral and halachic obligation to save another's life? The question is far too broad and nuanced to even discuss in this forum. I only want to point out a not well known paragraph from Reb Meir Simcha in his Ohr Samei'ach on this topic. Every word in Reb Meir Simcha is a gem, and this one is particularly interesting and deserves wider dissemination.
The issue at hand is whether one is allowed to endanger his life to save someone else. Assuming he is allowed to do so, is he obligated to do so? Does the relative degree of danger to himself and the other person matter? There has to be some leeway here; an absolute prohibition would mean there were no soldiers, no policemen, no doctors in infectious disease clinics, and no firefighters. As you can see, the numerous permutations require careful attention. One current question is whether a person may donate a kidney or part of a liver to someone who would otherwise die. There is definitely a danger involved in living donation. It is allowed? Is it justified? Is it a mitzvah? Is it an obligation? For helpful overviews and a survey of the various opinions, both modern and ancient, see here and here. There is a wonderful group, called HODS- the Halachic Organ Donor Society- that has a very interesting and informative web site dedicated to this and many other issues. I should really have linked to HODS' home page, but the video on the home page is so hard to watch, and so beautiful and poignant at the same time, that I wanted to warn you before linking to it.
As always, halachic questions of this nature should be directed to an Orthodox Rabbi of stature. Basic rabbinic training certainly does not guarantee the expertise to deal with such questions.
Back to Reb Meir Simcha. The Gemara says that a person who must run to the Ir Miklat, that is, a person who had killed someone inadvertently, may never leave until the Kohen Gadol dies. There is absolutely no circumstance that would allow him to leave. As the Mishna in Makkos 11b says,
Reb Meir Simcha addresses the question, how can it be that Pikuach Nefesh wouldn't allow temporary abatement of the punishment of Galus? If he is needed to lead the army in war, if he is needed to testify in a capital case, if he is a doctor, nothing at all allows him out, despite the deaths that will likely result from this refusal. What happened to the primacy of human life? In fact, I would say that the best possible thing for this man to do that would rectify his negligent taking of a life is to go and save lives! Why doesn't the Torah allow this?
Reb Meir Simcha (in 7 Rotzei'ach 8) answers that all these reasons would not affect the vengeful relative's right to kill him. The Goel Hadam will still have the right to kill this man as soon as he walks out of the Ir Miklat. Since leaving endangers him, he is not allowed to leave his safe refuge; A man may not endanger his life in order to save others. (I would not be too quick to assume that this is based on Reb Akiva in Bava Metzia 62a, חייך קודמין, because Reb Meir Simcha makes no reference to that Gemara at all.) This is particularly interesting in light of the fact that the ability and the motivation of the Goel Hadam to do what he is allowed to do will vary with the time and place and circumstances. But Reb Meir Simcha says that the mere fact that the Goel Hadam has the right to kill this man with impunity prohibits this man from leaving the Ir Miklat, even if leaving would enable him to save another man's life.
Without in any way deprecating Reb Meirs Simcha's teretz, the fact remains that, as he himself cites extensively, many poskim say that one may, perhaps even must, endanger himself to save another's life, and his teretz is of no use in explaining how the others would learn the Mishna in Makkos. (Rav Shternbuch, in Taam VaDaas, argues that the whole concept of war means that you endanger yourself for the benefit of the people, and it doesn't make any sense to say that it is the danger of the Goel HaDam that precludes a general from leaving the Ir Miklat in order to lead his army in war. I'm not impressed with his raya, but there it is.) A certain Isha Chashuva immediately answered that the experience of Galus has to be a simulacrum of death, of leaving everything behind and not being able to return. If the person can come back for any reason at all, then it's not at all like being dead. (I once said that the reason women tend to get headaches is because vestigial organs are more prone to derangement. While I still think this is true, my rebbitzen has made it clear that it does not apply to her.)
Here is our teretz.
Galus is a form of chiyuv missah. Chiyuv missah obviously trumps V'chai bahem. Proof- first of all, the fact that you kill the guy is proof enough. More- pikuach nefesh is docheh shabbos, but a mechallel Shabbos is chayav missah. One might say that this is because Rachmana afkerei l'damei. Or one might say that the chiyuv missa is doche pikuach nefesh. If you say the latter, then you have a pshat in the Mishna in Makkos. In any case, one thing is certain: you don't say אז מען דארף דעם גנב - נעמט מען אים פון די תליה אראפ