To what degree do humans have an ethical responsibility to sustain the species? Let's imagine a situation in which every single person on the planet decided to opt for voluntary sterilization (or every person of child-bearing age). Would this be unethical? Does the human species, as a species, have a responsibility to reproduce itself? Clearly, the planet and the other species on it would, on balance, be much better off without humans on it.

This is a fascinating question, in some ways, I think, it's connected to the question of whether we have responsibilities to things bigger than us in the sense of things that can continue to exist without us--e.g. things like families, nations, political and artistic movements, cultures, universities, businesses. Generally, I would say there is a moral obligation to sustain good things generally--at least where sustaining some good thing doesn't require undermining another of greater value.

To your specific question: I would say that considering the species itself, there is no obligation to sustain it, since the simple existence of the species is neither good nor bad, so long as there are other species to substitute for it. Considering the species as part of the larger ecosystems of the planet, where genetic diversity is salutary for the health of living things generally, there is an obligation insofar as the well being of living things is good. Considering the existence of the species as the necessary condition for all kinds of other goods, such as art, science, scholarship, virtue, culture, family, sport, etc., implies that an additional obligation exists as well. Of course, this last bit does depend upon the idea that the goods that the existence of human beings makes possible are not overwhelmed by the bad things their existence produces.

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