This post is inspired by the (public domain) story "Warrior Race" by Robert Sheckley (of whom I am a big fan).
Suppose I want a hundred dollars from you, but have no claim on it. So I resolve to torture myself in a way that would have significantly more disvalue than whatever good you can do with a hundred dollars on the condition that you don't give me the money, and convince you of my resolve. I also ensure you have no way of stopping me except by paying up. Or perhaps, if you're not sure of my resolve, I set up a machine that will torture me until you pay up. I also convince you that (a) I won't do this again, and (b) I will ensure nobody will ever find out about it. (If there are worries about the epistemic appropriateness of your trusting me, suppose that I have a little device implanted in my brain which will kill me if I am about to violate these rules.)
If you're a consistent utilitarian, you will pay up. Utilitarians, thus, are open to this particularly odd sort of blackmail.[note 1] Intuitively, I think, there is no duty for you to pay up. You could just say: "You made your bed, now lie in it." And so this is an argument against utilitarianism.
But why is it that non-utilitarians don't have to pay up? After all, it seems plausible independently of utilitarianism that if a moderate expenditure can prevent an immense amount of suffering, one has a duty to do that.
Or if that's not right, other forms of threat might work. You wanted to vote against Smith's getting tenure. But Smith informs you that if you vote against his tenure, he'll literally torture himself for the rest of his life to an intensity far disproportionate to the values involved in a fair tenure process. It is plausible that something like the proportionality condition from the Principle of Double Effect is a necessary condition on the permissibility of an action with a foreseen bad effect: the bad effect cannot be disproportionate to the good effect. But here the bad effect seems to be disproportionate to the good effect. (If causation doesn't filter through others' decisions, then suppose Smith set up a machine to torture him if you vote against him.) If this is right, then we don't have an argument against utilitarianism. We just have the observation that threats of self-harm will be effective against virtuous people.
One might think that anybody who would issue such threats of self-harm is insane, and maybe it is not so implausible to suppose that an insane person could get you to do whatever (within very broad limits) she wants by means of threats of self-harm. But if you're known to consistently act by a moral theory, like utiltiarianism, that requires you to give in to the demand, then it is not insane to threaten self-harm in this way, as the threatener knows that she won't have to carry out the threat. It can, indeed, be narrowly self-interestedly rational.
I think there may be a move available to the non-utilitarian. She could insist that your suffering the torture involves goods of justice. There are (at least) two kinds of punishment: imposed and natural. And justice is involved with both. It does seem plausible that if two people are drowning, and only one can be rescued, and one is there because she murderously pushed the other in and in the process toppled in with her victim, the innocent has a call on us that the other does not.
Notice, though, that in these sorts of cases as individuals we have no right to impose a punishment on the person other than public disapproval. As individuals certainly we have no right to impose torture on someone who threatens self-harm and no right to impose death on the drowning attempted-murderer. So if the right story involves natural punishment, and "You made your bed..." suggests that, then we will still need a doing/non-doing or foreseeing/intending distinction. Actually, doing/non-doing won't work in the tenure case, since there Smith threatens you with self-harm if you vote against him, and voting is a doing. So it seems one needs a foreseeing/intending distinction to make this work out: you foresee that Smith will suffer, but because the suffering would be a good of justice, that shouldn't sway you from your vote against him.[note 2]
Furthermore, the concept of punishment without a punisher appears incoherent. So to make the "natural punishment" line go through, one may need a God behind nature. Maybe one could try for "natural consequences" that aren't punishment. But if they aren't punishment, it's not clear how the threatener's suffering the torments is a good of justice. If they aren't punishment, all we can get is that it's not unjust that the threatener should suffer. But that could leave intact the argument that you shouldn't vote in such a way that will cause this disproportionate suffering as, plausibly, that wasn't an argument from justice but from non-maleficence.
So it could well be that supporting the "You made your bed..." line in these cases requires fair amount of philosophical doctrine: justice and natural punishment, foreseeing/intending and maybe even theism.
Of course, it could be that the hard-nosed "You made your bed..." intuitions are wrong.