Now, I'm still not convinced that Dumbledore even has an opposite at all, but I think the most likely candidates within the Potterverse are Grindelwald and Aberforth. Grindelwald because he's probably the character in the book who comes closest to Dumbledore's level of brilliance but lacks his capacity for compassion. I'm not sure about that, though, because Harry never actually meets Grindelwald in person, so everything that we know about him comes to us secondhand. Then Aberforth is a candidate because he's so practical and concrete where Albus tends to be more theoretical and abstract.
Now, outside the Potterverse, one that I thought of right away but didn't bother to justify to myself was Darth Vader, for which eofs did a lovely, detailed analysis, which is very thought-provoking. But I think, really, if forced to choose a character that was the opposite of Dumbledore, I would go with Gandalf, because where Dumbledore is warm and nurturing, Gandalf is cool and stand-offish. Also, Gandalf comes back to life for no good or explicable reason; whereas Dumbledore, while he does communicate from beyond the grave, has the decency to stay dead and not toy with our emotions.
The reason why I asked myself the question in the first place is because it is my intention to enter this contest, and I started off by making a list of things that I wanted to accomplish in any stor(ies) that I wrote set in the Dark Crystal universe, and near the top of my list was that I wanted to make the Mystics more interesting characters, because they don't really get to do much in the movie. They have informed personalities, but apart from some brief exposition, all they really do in the movie is chant a little and lumber around a lot. (Side note: since working for the orthopedic clinic, I have to concentrate hard in order to write "lumber" instead of "lumbar," but I'm glad it's not the other way around.)
But that caused a problem, because how does one develop characters whose passivity is not only central to their characterization but also to the main conflict of the story? To solve the problem, I looked to other authors I admire, including J.K. Rowling. I thought about Dumbledore, who is very powerful, able and willing to engage an enemy in battle if need be; but at the same time, like any good teacher, he knows that his students won't learn anything if he just gives them the answers, so more often than not he stays in the background and observes, only occasionally intervening to give information when someone gets stuck, or to serve as the last line of defense when someone gets cornered by Death Eaters.
And that helped me a lot. Suddenly I understood that the Mystics in my stor(ies) must serve in a mentoring capacity. They have to teach because they literally can't do. And then, with a cheerful "Have fun storming the Castle!", they send Our Hero(es) off on their way. In other words, the Mystics are a race of Dumbledores: four-armed, long-necked, vaguely saurian Dumbledores.
Now, for those of you not familiar with The Dark Crystal, the antagonists are the Skeksis (the singular is the same as the plural), who are the opposite of the Mystics. So that made me wonder, "If the Mystics are a race of Dumbledores, what does that make the Skeksis? What is the opposite of Dumbledore?" And it's really nothing more than an intellectual type of question that I answer for my own amusement. It's not going to have an impact on the way I write my stor(ies) because the Skeksis are characterized pretty strongly in the movie and I never really had any question about what I was going to do with them or how I would protray them.
Incidentally, the more I think about it, the more I think that if I were going to compare the Skeksis to any other character, it would be Voldemort. Like Voldemort, they're ruthless and grotesque, and their obsessions include ruling the world and preventing themselves from dying (at the expense of others). So the Mystics are like a race of Dumbledores and the Skeksis are like a race of Voldemorts. And while I tend to think of Voldemort more as the opposite of Harry, there is certainly a case to be made for Voldemort as the opposite of Dumbledore ("the only one he ever feared" and all that stuff).
So, that all makes sense to me now. Clearly, I just started off by asking myself the wrong question.