Odysseus is told by Athena that he is very unlikely to reach Ithaca, unless he suppresses his desire to reach Ithaca. If he does suppress it, he will quickly by accident find his way to Ithaca, and as soon as he is within ten stadia of it, his desire will return. Athena points to a pear from a tree on the banks of Lethe, and tells him that this pear will suppress his desire to reach Ithaca. But Odysseus longs for Ithaca too much to be willing to let go of his desire to return there. He tosses the pear by the wayside and wanders the world for many years.
Odysseus was irrational to hold on to his desire to return to Ithaca. It was thereafter irrational for him to have the desire. Yet the desire to return to his home was a perfectly rational one.
Irene never desired to experience friendship. Finally, one day, Matthew gave her a fallacious argument whose conclusion was that friendship is worth having. Irene didn't see the fallacy, and concluded that friendship is worth having. She then hired Dr. Mesmer to hypnotize her into desiring friendship. A couple of months later, while having an intense desire for friendship, she found the fallacy in Matthew's argument. But while she believes that friendship is not worth having or desiring, she irrationally refuses to hire Dr. Memser to hypnotize the desire for friendship away.
Irene acquired her desire for friendship irrationally and is irrational in holding on to the desire. But the desire for friendship is perfect rational.
Patrick comes to be convinced by Irene, whom he has excellent reason to think to be an epistemic authority even whe she says things that seem absurd (she has said many seemingly absurd things to him, and turned out to be right), that he ought to desire to be the heaviest man on earth. By constantly dwelling on the excellent reasons he has for trusting Irene, and on Irene's advice, he comes to desire to be the heaviest man on earth, and starts to eat.
Patrick acquired his desire to be the heaviest man on earth quite rationally, and is rational in holding on to the desire. But the desire is irrational.
Collectively, the cases force a distinction between (ir)rationally having or acquiring a desire, and a desire being itself (ir)rational.
But now what does the irrationality or rationality of a desire in itself consist in if it does not consist in the irrationality or rationality of the agent who has it in respect of the having of the desire? I suspect that a good answer will have to advert to the human good, to human flourishing, but even so, I don't know how to answer.