is it always rational to make others believe things that are true?

This depends on your values. If chief among them is "honesty", and you caveat the "make others believe things that are true" with a "for the right reasons" clause, then probably, yeah. If honesty has to compete with things like keeping your property, maybe not.

If I am playing the Prisoner's Dilemma with someone (just once, so no tit-for-tat or anything), and I have the choice of making my opponent either act rationally or irrationally, the rational thing for me to do is make him act irrationally.

I'm not sure what the content of "making your opponent behave (ir)rationally" is supposed to be. It's certainly not an uncontroversial tidbit of received wisdom that the rational thing to do in the Prisoner's Dilemma is to defect, which is what you seem to imply.

We think that religions are false, and a shared priority of Less Wrong denizens is to believe things that are true instead. I'll readily admit that religion has some good effects. Many people find it comforting; it's inspired great works of art and music and architecture; it does a lot of work to funnel money to charitable causes, some of which are very helpful; it encourages community-building; and it has historically served as a cultural touchstone to enable the development of some very powerful iconography and tropes.

It's still false.

If you like the good things about religion, there are alternatives (although most of them only work piecemeal). For instance, there's Ethical Culture, which fills in the community gap a departing religion can leave.

Make everyone rational, and they will do the rational thing, i.e. defect when faced with a prisoner's dilemma (or tragedy of the commons, or whatever).

You assume too much.

Also, have you spent any time searching for a third alternative before deciding that religion is the only thing that can keep people from destroying each other?

I would dispute the (implied) part of this post that suggests removing religion would necessarily be a good thing. Besides being a way to explain rainbows and earthquakes and whatnot, it is also a "solution" to the prisoner's dilemma. For explanations of physical phenomena, religion is no longer needed. But the "morality" problem is still there. Make everyone rational, and they will do the rational thing, i.e. defect when faced with a prisoner's dilemma (or tragedy of the commons, or whatever). Getting rid of religion may certainly have its benefits, but I would not be too sure it would be an overall good thing.

[This is not a quote, but a meta discussion.]

I find it curious that the quotes posted here have higher votes on average than the usual discussion comments, and it makes me think that I have a below-average appreciation for quotations. Why do people value them, I wonder?

I have met people who exaggerate the differences [between the morality of different cultures], because they have not distinguished between differences of morality and differences of belief about facts. For example, one man said to me, "Three hundred years ago people in England were putting witches to death. Was that what you call the Rule of Human Nature or Right Conduct?" But surely the reason we do not execute witches is that we do not believe there are such things. If we did-if we really thought that there were people going about who had sold themselves to the devil and received supernatural powers from him in return and were using these powers to kill their neighbours or drive them mad or bring bad weather, surely we would all agree that if anyone deserved the death penalty, then these filthy quislings did. There is no difference of moral principle here: the difference is simply about matter of fact. It may be a great advance in knowledge not to believe in witches: there is no moral advance in not executing them when you do not think they are there. You would not call a man humane for ceasing to set mousetraps if he did so because he believed there were no mice in the house.

-C.S. Lewis

Kasparov competed against Deep Blue to steer the chessboard into a region where he won - knights and bishops were only his pawns

Were you trying to mix the literal and metaphorical here? Because I think that just his pawns were his pawns :)