Why is it philosophically false to point out the errors in false ideas?

Criticizing religion is not philosophically false, and it certainly needs to be done. However, attacks on religion alone, without a rational alternative offered, will be limited in their cultural impact. The reason stems from the central role of values in human life, and religion's historical role in having a near-monopoly on that subject.

Many of the popular critics of religion are skeptics who offer nothing other than individual whims or arbitrary democratic consensus as a guide to human actions. Many people know that such theories of ethics would be disastrous, even if they cannot articulate why. They will choose a theory that they regard as vital to human life, even if illogical, over one that seems supported by abstract reasoning but would lead to death and misery in practice. (The argument that religious morality has clear faults will not be persuasive if people regard secular morality as unavoidably worse, such as not being able to prohibit murder.)

The argument above is not a concession to people's existing irrationality. People do have responsibility for their own ideas, including compartmentalization and other evasions. However, the fact is that most people are not - and never will be - professional intellectuals, and they cannot be expected to examine in detail all of the ideas promulgated by philosophers and religious leaders, let alone all of the implicit philosophical assumptions (such as the theory-practice dichotomy) that are nearly unquestioned in our culture. To them, life-or-death ethical questions such as, "Is it wrong to commit murder?" are naturally going to have more salience than metaphysical ones such as, "Does the universe need an explanation?" If faced with what appears to be a conflict between the two, non-intellectuals will naturally choose a worldview that seems to give a satisfactory answer to questions about human choices.

The task of Objectivists is to break the tyranny of this "noble lie" in people's minds. We need to show that there exists a philosophy that is both secular and life-supporting; and in fact, that the two must go together.

I agree with the questioner that there is something pernicious in dismissing criticism as useless. Even beyond the fact that you aren't going to change anyone's religious loyalties without raising the spectre of serious inconsistencies and inhuman implications in the doctrine they accept, pointing out the irrationality of a set of beliefs sets up cognitive dissonance--intellectual feelings that something isn't right--which lessens the believer's devotion and confidence.

There are traditions of humanistic thinking that aren't fundamentally religious, and the Classical, Renaissance, and Enlightenment movements exhibit their power. I do not see why well-reasoned attacks on religious doctrine or practices are or would be expected to be "inherenty limited." Objectivism doesn't aim to slip a different set of beliefs into the heads of unthinking people. It will be necessary for people to think their way out of religion before they can think their way into Objectivism.

It is not philosophically false to point out the errors in someone else's argument. It is however incomplete. To be complete you need to show the errors in someone else's argument, while providing a correct argument.