The history of science is replete with discoveries that were considered socially, morally, or emotionally dangerous in their time; the Copernican and Darwinian revolutions are the most obvious. What is your dangerous idea? An idea you think about (not necessarily one you originated) that is dangerous not because it is assumed to be false, but because it might be true?
I find this a rather absurd sort of question; a serious look at almost any answer that might be given to a question like this will make clear to the impartial observer that all the ideas are actually fairly tame (and, indeed, in most cases weren't even new), and most of the revolutions are not in the ideas (even assumed true) but in the means of showing that they are, in fact, true. What's dangerous instead is the hyperbole used to make them sound dangerous. One finds this in Dennett, for instance; the theory of natural selection, in its basic scientific formulation, is actually not particularly dangerous to anything. If half the poorly-supported and extreme claims Dennett makes for it were true, it would, indeed, be the 'universal acid' Dennett claimed it was; but, in fact, the claims rest on nothing but Dennett's very vague, purely imaginative, and inadequately analyzed sense of analogy between the theory of natural selection and various metaphysical positions. I have complained of this analogical leap, which is extremely common, before. The answers to The Edge's question serve as a reminder that in matters of science and fundamental questions we should never rest with the vague sense of analogy but demand serious analysis. We would never accept an inference like the following as legitimate:
Bats flap around; fish out of water flap around; therefore bats are fish out of water.
And yet the ability of reasonable people to accept without any critical thought similar inferences (or inferences that would turn out to be similar if anyone bothered to work them out carefully) when science is involved is utterly astounding -- and, needless to say, disturbing. Scientific discoveries should not be ways of avoiding serious rational analysis; they should be sparks inspiring it. But the desire to take shortcuts prevails, and we get the philosophical equivalent of quackery and snake oil. Give people a scientific discovery and they start making wild philosophical claims, without the slightest recognition that they are doing exactly the same thing Eddington did when he claimed that modern physics proved idealism (and more culpably, too, since Eddington was a serious enough thinker that he eventually caught his mistake, whereas many of these never do). The answers to The Edge's question show this in spades. We have Pizarro's answer, which confuses together the positions of moral realism, moral sense theory, and moral infallibilism (one can be a moral realist but not a moral sense theorist, or a moral sense theorist but not a moral realist, or both, or neither; and one can be either without believing that there is a 'royal road to moral truth'); or Anderson's bizarre argument about the probability of God's existence; or Kosslyn's odd talk about God and Supersets; or any number of others. We need rational thought, not mush. (Of course, it's easy to criticize scientists for the trait; but one can hardly expect otherwise. Not every scientist has the philosophical acumen of a Darwin or a Planck. And when ordinary people do it, it is still unfortunate, but it is an entirely understandable mistake. The real shame is when philosophers do it; and it is, alas, no less common there.)
But it must also be said that there's hope, since some of the answers are actually pretty good, given the absurdity of the question. (Like Chris, I really enjoyed Gopnik's, for instance.)
[Incidentally, why is it that everyone thinks that the revolutionary aspect of the Copernican revolution was that it removed the earth from the center of the universe? The real revolution, it has always seemed to me, which was due more to Galileo's Copernicanism than to Copernicus himself, was in demolishing the distinction between terrestrial and celestial matter. With that distinction, something like Aristotelian cosmology is a necessary truth; that distinction gone, Aristotelian cosmology collapses completely. Sir John Herschel, if I remember correctly, recognizes this in his Preliminary Discourse; but it's a point that should be more generally considered. It seems to me that the primary scientific significance of removing the earth from the center was at the time just one of measurement; which is perhaps why instrumentalism about Copernicus's theory was so often considered reasonable. Just a thought.]