Here is a story I’ve heard from mathematicians in a few different contexts. I will quote it from Stilwell’s text Four Pillars of Geometry.
“The first theorems of hyperbolic geometry were derived by the Italian Jesuit Girolamo Saccheri in an attempt to prove the parallel axiom. In his 1773 book, Euclides ab omni naevo vindicatus (Euclid cleared of every flaw), Saccheri assumed the non-Euclidean parallel hypothesis, and sought a contradiction. What he found were asymptotic lines: lines that do not meet but approach each other arbitrarily closely. This discovery was curious, and more curious at infinity, where Saccheri claimed the asymptotic lines would meet and have a common perpendicular. Finding this ‘repugnant to the nature of a straight line,’ he declared victory for Euclid.”
I’ve heard this story a few times, and each time it was to illustrate the importance of deriving a contradiction in a reductio, not just stopping when things get unintuitive or “repugnant”. I haven’t seen the original book, so I don’t know if Saccheri says that his conclusions contradict the nature of a straight line, in which case he would have misunderstood what was happening to the notion of line under the new assumption. However, if he thinks that the repugnance is enough for a reductio, he is wrong.
(Apparently there is a lot of literature on intuitions. I’ve read none of it. It could very well discuss this story in excruciating detail.)

It seems, just from my own reading, that the charge of having unintuitive consequences is often leveled against various theses. I don’t have a nice example off-hand, but I’m doubtful that it will be terribly hard to track one down. (I think I used the phrase in the paper I read at UT Austin, but I am pretty sure I have seen real philosophers use it.) It is never clear to me what the force of that charge is. If it is meant in the sense that Saccheri supposedly used it, then that seems to be bad. If it is meant ad hominem, then it is fairly easy to ignore. There are certainly at least some areas of philosophy, logic at least, where the charge of unintuitive consequences is fairly meaningless. Are there any areas in which saying that something has unintuitive consequences is particularly damning? There probably are, but I’m not sure what exactly.