Let us put all emotion aside, then. Let us look at the solid facts of the situation, from that determine the moral status of the various parties, and thus finally the moral realities which must be followed. 1, 2, 3. OK?
We are going to do this for the following situation: a mother has given birth 1 month ago to a baby that she and the father no longer wish to care for.
Solid facts:
1. The baby is a parasite upon the mother and father. They are supporting it.
2. The baby cannot support itself on its own.
3. If the parents remove their support, immediately evict the baby from their home, placing it in the waste receptacle, and do nothing else, the baby will die in short order.
4. The home, food, and other resources which the parents are using to support the baby are their own just property.
5. The baby lacks many of the defining characteristics of a human and will not acquire them for quite some time.
Moral Stati:
6. The parents are humans, with full human rights over all their property -- their bodies, their time, their food, their shelter, etc.
7. The baby, lacking many human characteristics, cannot be considered a full human in the same sense as the parents and humans in general.
8. The baby thus does not fully possess human rights.
9. However, the baby does possess the potential to become fully human, and given time he probably will. This potential humanity may or may not change the baby's moral status in your view. We will leave aside this question for the time being because it is complex.
10. The parents up to now have been closely associated with the baby, but now wish to end that association. So up to now, the parasite-host relationship has been voluntary and no trespass nor infringement has occurred. That has changed, however, as of now.
Moral Realities:
11. Libertarianism, as you said, and I agree, does not as a rule impose positive obligations. That means that the parents cannot be forced with violence into supporting the unwanted baby forever as an "obligation."
12. Libertarianism also mandates free association. That means the freedom to associate or to not associate. Since the parents now want to not associate with the baby, they should be free to disassociate themselves.
13. Libertarianism supports property rights. The parents are the just owners of all the the property involved -- diapers, breast milk, crib, house... -- except for the infant's body, which due to the unresolved ambiguity of #9, is of unknown status. The parents thus have the right to make decisions over all of the property, except, again, for the body and person of the infant itself.
14. The infant cannot really make choices in a human choice-making way as far as we know, or at least cannot make them known in a way that we understand. So the question of what the infant may justly do is meaningless. The infant cannot "do" anything, neither just nor unjust. The only question of justice then is what the parents may justly do, (and, if we want to complicate it, what other outside human actors may do in response).
So, what say you all (I wouldn't want to limit myself to osan)? What is the conclusion then? What may the parents justly do? May they take the initiative and actively kill the baby however they wish? Is that forbidden but they may passively kill it by placing it inconspicuously in the garbage or road-side and allowing nature to take its course? Or are both these courses forbidden by justice?