But it depends a whole lot on what we mean by 'morality'.
What I mean by "morality" is the part of normativity ("what you really ought, all things considered, to do") that has to do with values (as opposed to rationality).
I agree that it's going to take a lot of work to fully clarify our concepts. I might be able to assign a less remote probability to 'morality turns out to be impossible to carefully reason with' if you could give an example of a similarly complex human discourse that turned out in the past to be 'impossible to carefully reason with'.
In general, I'm not sure how to show a negative like "it's impossible to reason carefully about subject X", so the best I can do is exhibit some subject that people don't know how to reason carefully about and intuitively seems like it may be impossible to reason carefully about. Take the question, "Which sets really exist?" (Do large cardinals exist, for example?) Is this a convincing example to you of another subject that may be impossible to reason carefully about?
What do I mean by "morality isn't logical"? I mean in the same sense that mathematics is logical but literary criticism isn't: the "reasoning" we use to think about morality doesn't resemble logical reasoning. All systems of logic, that I'm aware of, have a concept of proof and a method of verifying with high degree of certainty whether an argument constitutes a proof. As long as the logic is consistent (and we have good reason to think that many of them are), once we verify a proof we can accept its conclusion without worrying that there may be another proof that makes the opposite conclusion. With morality though, we have no such method, and people all the time make moral arguments that can be reversed or called into question by other moral arguments. (Edit: For an example of this, see these posts.)
Without being a system of logic, moral philosophical reasoning likely (or at least plausibly) doesn't have any of the nice properties that a well-constructed system of logic would have, for example, consistency, validity, soundness, or even the more basic property that considering arguments in a different order, or in a different mood, won't cause a person to accept an entirely different set of conclusions. For all we know, somebody trying to reason about a moral concept like "fairness" may just be taking a random walk as they move from one conclusion to another based on moral arguments they encounter or think up.
In a recent post, Eliezer said "morality is logic", by which he seems to mean... well, I'm still not exactly sure what, but one interpretation is that a person's cognition about morality can be described as an algorithm, and that algorithm can be studied using logical reasoning. (Which of course is true, but in that sense both math and literary criticism as well as every other subject of human study would be logic.) In any case, I don't think Eliezer is explicitly claiming that an algorithm-for-thinking-about-morality constitutes an algorithm-for-doing-logic, but I worry that the characterization of "morality is logic" may cause some connotations of "logic" to be inappropriately sneaked into "morality". For example Eliezer seems to (at least at one point) assume that considering moral arguments in a different order won't cause a human to accept an entirely different set of conclusions, and maybe this is why. To fight this potential sneaking of connotations, I suggest that when you see the phrase "morality is logic", remind yourself that morality isn't logical.