At first glance, it might seem that being intellectually humble is at odds with having strong convictions on matters of controversy. (Here I will focus on the messy moral cases.) This is because (a) we might assume that at least some of the people with whom we disagree are, in some sense, our "epistemic peers" (that they share familiarity with the ins and outs, arguments, etc., bearing on the issue, and are just as reasonable, thoughtful, etc., as we are) and (b) an intellectually humble person will appreciate that the features that make those others one's epistemic peer imply that one is one among many, and that there is no prima facie reason to suppose oneself more likely to be right about the controversial issue. (For those familiar with the literature on the epistemology of disagreement, it might seem that intellectual humility favors adopting the Equal Weight View, which says that we should "split the difference" in cases of disagreement and move in the direction of agnosticism.)
The difficulty with this is that we may find ourselves strongly convinced of, and practically committed to, some controversial moral view because we still find the considerations in favor of that view the best. Is remaining unmoved irrational, or unreasonable? One way to avoid this is to conclude that those who disagree are not after all my epistemic peers. (Tom Kelly criticizes the Equal Weight View for making it too hard to ever arrive reasonably at this judgment once I've initially judged that a person is my peer. [Warning: it's a long paper!]) In a lot of cases, this might seem like a cheap move (though it's one, as Kelly notes, that Elga (above) seems to make, too, in explaining why the Equal Weight View doesn't have radical agnostic consequences in the non-ideal real world.)
Moral cases are particularly tricky because practical considerations are intertwined with "pure" epistemic (or theoretical) considerations, and as Catherine Elgin notes (in her essay in the disagreement volume linked above) that it may not be psychologically possible at any rate just to abandon (an especially deeply held) belief. I'm working on a line of thought that suggests a rough distinction between intellectual and practical contexts, and suggests that to be intellectually humble about our convictions, we have to moderate our acceptance of those convictions in contexts where the issue is up for (say, philosophical) discussion and debate. We can't just (obnoxiously) beg the question. Practically, we have strong reasons for accepting the view as a personal policy (in our own moral decision-making, etc.), since it is, after all, our conviction. (It's being our own doesn't give it special epistemic weight, but just special practical weight.) (By "acceptance" here, I mean it in the sense articulated by L. Jonathan Cohen, who says that to accept p is to take it as a premise in argument, deliberation, etc.) So, humility requires something like not brow-beating those who disagree with us, or being merely dogmatic. We have to try to discuss the issue as if it is open (even if we think it is not), for multiple reasons (both to attempt to persuade others and, I think, to show due respect for their not-yet-rejected status as peers).
But there can be cases where even this seems too demanding (or somehow alienating). A person's convictions might involve the thought that the alternative view is pretty bad, and that although there is a controversy about the issue, one thinks this itself shows something pretty bad, say, about the state of society. Perhaps, e.g., along the lines of what Gaita says about those who think that torture ought to be "undiscussable." Maybe if it turns out that there's just no arguing with the other side--or as Cavell says, "no hope of agreement"--then one is warranted in rejecting that those others are epistemic (or moral) peers. (And you go engage in activism and rhetoric and so forth, rather than argument of a more rational form.)
Something else that I think can be said here is that humility doesn't require moderating one's belief (or acceptance) in those cases where the opposing view is itself a product of a failure of humility (in the broader, moral sense). Take those who have struggled against various forms of oppression, racism, sexism, etc. Where they have good reason for thinking that those who disagree are motivated by a sense of moral superiority, then their views are themselves (I suggest) incompatible with humility, ruled out as immoral by it. One needn't humbly "split the difference" with those who are motivated (intentionally or not) by a kind of arrogance, self-righteousness, etc.
There's more to say, I think, but that's where I'm at. (The rest of the puzzle has to do with the cases where our convictions call for practical action that's directed at those with whom we disagree, and what we might say about that from the perspective of humility.)