Ben Golub, Economics Ph.D candidate at Stanford [GSB]
Rand argued in The Virtue of Selfishness, her nonfiction work on moral philosophy, that (i) one's own life is logically the ultimate value because it makes all other values possible; that (ii) it is therefore irrational for a valuing being not to defend and further this life above all other values; and that (iii) this entails strong conclusions about the rightness -- actually, the moral necessity -- of living selfishly. More broadly, a central theme in her work is the proposition that a particular conception of the good life is itself a moral value, and that that from this value, certain libertarian moral and political conclusions follow.
Influential philosophical analyses of Rand's thinking hold that the argument above is not a sound derivation of the moral rightness of selfishness. In particular, Nozick (who was sympathetic to Rand's libertarian position) pointed out that the move from step (i) to step (ii) is a mistake. It confuses (a) the fact that is physically necessary for a person to remain alive in order for him to continue to have values with (b) themoral priority of his own life over the other things he (or others) might value. In other words, if John sacrifices himself now for something he now values (such as the end of a war), then after death he might not continue valuing the peace he sacrificed himself to achieve. However, that does not mean that he is negating the value of peace (to his current self) by achieving it. He is merely destroying the possibility that a future John could be valuing and enjoying peace. And that is quite different.
To get around this objection and make Rand's argument work -- to show that John is irrational -- one needs to assume that the only things John is rationally allowed to value are those things that involve him continuing to live and value them in the future. But that already assumes that self-sacrifice is irrational, and that was what the argument was supposed to prove, not assume. More broadly, many of Rand's moral arguments have this flavor: they purport to deduce that a rational being must behave in a certain way to have a good life, but all the conclusions about morality are smuggled in through a morally loaded definition of rationality, or of human nature.
Ironically, the premises of Rand's philosophical argument are sometimes in tension with the romantic vision articulated in her novels, in which heroes are willing to sacrifice themselves for the sake of integrity in a hopeless world.
Of Rand's political theory, Kukathas says that "[i]ts unremitting hostility towards the state and taxation sits inconsistently with a rejection of anarchism, and her attempts to resolve the difficulty are ill-thought out and unsystematic", rendering her work on this matter "of little interest" to political philosophers.
It is certainly hard to say whether Rand was abstractly "wrong" in her belief that it is very immoral to require certain people to sacrifice valuable things for the benefit of others, as happens in income redistribution. People of different political positions disagree about this, but many hold that this is a dispute about fundamental values. If the mainstream philosophical consensus is to be believed, Rand was wrong to claim that this dispute can be resolved objectively by an appeal to reasoned argument from uncontroversial axioms.
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A note on sources: this answer draws heavily on two excellent published articles on this question -- one short, one long. The short one is by Chandran Kukathas, who gives a remarkably sensitive and nuanced -- though ultimately critical -- view of Rand in his article on her in the Routledge Encyclopedia of Philosophy (http://www.rep.routledge.com/art..., subscription required).
The long one is the article "On the Randian Argument" by the renowned libertarian philosopher Robert Nozick, which can be found in his book Socratic Puzzles.
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