The more conventional opponents of intentionalism are formalists, who believe that the meaning of a text can be determined by the words of the text itself, without recourse to things like the author's intention. (The New Critics were the paradigmatic formalists.) There are also exotic views like reader-response theory, according to which the reader or the interpretive community determines what the text means; and deconstructionism, which seems to hold that texts don't have unified meanings. What I'll try to do here is explain why intentionalism is the best theory of interpretation. I'm going to direct my attacks mostly at formalism here, because it strikes me as the strongest opponent.
First, a simple example. Suppose someone tells you, "I went to the bank today." This could mean either that they went to the side of a river, or that they went to a financial institution. In some circumstances, there won't be enough context to rule in favor of one interpretation or the other. (The fact that the person actually went to a financial institution and not the side of a river doesn't make it true that they meant that they went to a financial institution, although it's good evidence for that -- they still might have meant that they went to the riverside, and been lying. But the reason it is good evidence is that it makes it more probable that they intended to communicate that they went to a financial institution.) Even if there's not much context, we still want to say that they meant one thing or the other. It's hard to see what could make it the case that they meant one thing or the other in cases where there isn't a lot of context, unless it's their intention that did it.
I also don't know how a formalist makes sense of our judgments about which literary works are satirical. If we found something like A Modest Proposal coming from a society that practiced cannibalism, we might be tempted to read it as an earnest suggestion, as the intention to promote cannibalism is one that a cannibal might have. But our understanding of the sorts of things that literate Irishmen of Jonathan Swift's times were likely to regard as acceptable makes it wildly implausible to ascribe pro-cannibalistic intentions to him. So we regard his work as satirical. The best way to explain our different responses to Swift's work and to the identical words, written by a cannibal, is to go to our judgments about what the two authors are likely to intend.
I've aimed both of these examples at the formalists, but they work against reader-response theorists and deconstructionists too. Even if a reader, or an entire interpretive community, believed that Swift was earnestly proposing cannibalism, they'd be wrong. And it's hard for me to see any support for the idea that texts don't have unified meanings in these cases -- in Swift's case, we insist on a particular unified meaning -- the satirical one.
Now I'll deal with a case that's supposed to cause intentionalists trouble. Amanda posted a while ago about how Ray Bradbury has told us that Fahrenheit 451 isn't about censorship. According to him, it's "a story about how television destroys interest in reading literature." I haven't gone back and read the book to see if this actually makes sense, but in any case it's not unheard of for authors to make statements about the true meaning of their works that seem ridiculous. Does this mean that we were, in fact, all wrong about what their works meant?
Not necessarily. The causes of our actions often differ from the way we rationalize them, and in complex cases of communication like the writing of novels, people might not have a good handle on the psychological processes motivating them to write exactly as they do. So even as the forces within them cause them to focus their writing in a specific fashion and to play up aspects of stories that emotionally resonate with them, they may not be able to give correct descriptions of what's going on inside them. Perhaps what really drove Bradbury to write the story as he did -- and what he would've felt strangely unsatisfied if he hadn't been able to properly express -- was an appreciation of how bad government censorship of literature was, even though he didn't explicitly realize that this was motivating his portrayals of events and characters. I'm willing to give a fair amount of credence to first-person reports of one's intention, but even honest people aren't perfect at reporting their intentions in complex cases, so their reports can be overridden if we have enough contrary data.
Another case, brought to me by English grad students in Dr. Martinich's seminar five years ago, concerned the poetry of Arthur Rimbaud. They explained to me that some of Rimbaud's poetry seems to leave some aspects of its meaning open for the reader to fill in. This made reader-response theory seem more plausible to them. It did concern them, though, that not all interpretations of the poems they described would be correct, even if every reader had them. For example, Vowels is not a casserole recipe, no matter what a casserole-obsessed interpretive community might say. There's no good way to fill in the ambiguities in the poem and get there.
I don't know if they eventually accepted it, but they were intrigued by my way of reconciling their interpretive practice with intentionalism. Suppose Rimbaud intended aspects of the meaning of his poems to be filled in by his readers, so that it would mean different things depending on who was reading it. And suppose the readers took seriously the project of operating within the space generated by Rimbaud's intentions and did not try to turn the poems into casserole recipes. Then the meaning of his poems would be generated by something akin to an act of joint authorship. When two people's intentions are in harmony, they can jointly generate the meaning of a text, and to my English department friends, this seemed to be at least a plausible story about how one reads Rimbaud.
I should say a last word about the practice of reappropriating texts. In taking ahold of a text -- perhaps a play or a novel -- and giving it a meaning that you know the author didn't intend, either in your own mind or in some work you create based on it, you cease to be an interpreter and start being a kind of author. This can be good or bad, depending on how good an author you are, but the important point is that the responsibility for the things you intended but the original author didn't goes to you. If you take it upon yourself to turn Vowels into a casserole recipe, don't blame Rimbaud if it the casserole doesn't taste good. But if it does, most of the credit belongs to you, and not to Rimbaud.