Do I have a moral obligation to help others, even at the cost of sacrificing my own well-being, time, resources, health, etc.?
The answer is debatable, but in my opinion, no, I am not obligated to sacrifice anything of my own in order to help another, without that other person giving something in return.
Note that charity, the willing giving of resources with no expectation of reward, is still a good thing. But what you are claiming is that I should be *required* to allow my empty house to be lived in by a homeless person.
Forced charity isn't charity, it is theft. You have no right to demand anything from me, not my food, not my empty house, not my protection, none of it. If I give it to you freely, count your blessings, but the second you cross over into entitlement ("I deserve this and that, and if you refuse to give it to me, you're an asshole"), you've just earned yourself a giant "Fuck off."
That is what the NAP is, at it's core: you provide mutual benefit, or you fuck the hell off.