Non-Christians are not obligated to fix Christianity
The first problem with this argument is that failing to disabuse your friend of their incorrect Biblical views is not the same as endorsing them. Clark's ultimate goal is to attack secular intellectuals like Neil DeGrasse Tyson who have (falsely) affirmatively endorsed the idea that the Bible teaches young-earth creationism, while (correctly) maintaining that young-earth creationism is wrong. I agree that Tyson and others shouldn't take this position about the Bible (for reasons explained below). But that doesn't mean that someone who challenges the empirical correctness of a wrong interpretation of the Bible is thereby endorsing said interpretation. Our non-Christian Ohio-believer could simply remain neutral on the question of what the Bible teaches about Ohio's existence.
Second, Clark's argument assumes that all falsehoods are equal. He focuses on counting up the total number of falsehoods believed. But of course falsehoods differ in quality as well as quantity. Some falsehoods are farther from the truth than others -- compare "Ohio doesn't exist" to "Ohio is located in the South." And some falsehoods are more relevant to our lives (and thereby more damaging to believe) than others -- compare "Ohio doesn't exist" to "the state of Xhix on the planet Xarth orbiting the star Zeta Reticuli doesn't exist." It is perfectly rational to focus one's energy on the bigger, more consequential falsehoods.
But third and most important, Clark's argument assumes that a non-Christian can engage in debate over the correctness of a Biblical interpretation. Here Clark's non-Christian who insists Matthew teaches nothing on the subject of Ohio's existence is making the same mistake as Tyson is when he claims the Bible teaches young-earth creationism. The Bible (like any other complex text) is not self-interpreting. There is no single meaning that can be proven to be correct through an unimpeachable reading technique. This is a point that Clark himself has repeatedly stressed on his blog. In lambasting Biblical "literalists" (such as the creationists that the post under consideration is aimed at!), he emphasizes that the Bible always requires interpretation and that its meaning cannot be discerned by any simple reading technique. This fact is obscured in the Ohio example because he picked for his Biblical falsehood something so obviously absurd that it's difficult to imagine any realistic hermeneutic that could lead someone to believe that the bible teaches that. But that's certainly not the case for topics like creationism (or gay rights, or feminism, or any other topic where a liberal like Clark would disagree with common assertions about what the Bible says).
In order to determine which reading of the Bible -- the Ohio-affirming or the Ohio-denying, the creationist or the evolution-compatible, the anti-gay or the pro-gay -- is true and correct, one requires faith. I don't mean faith in the sense that God's spirit must guide your reading (though many Christians do believe in such a thing). I mean that if you don't believe in the existence of the Christian god, then it is meaningless to assert that there is a correct and an incorrect interpretation of the Bible. The correct interpretation of the Bible can mean nothing other than the one that God wants his followers to derive from the text. And so if one does not believe in the Christian God, one cannot believe there is a single correct interpretation of the Bible. In the same way that non-Christians can't be asked to distinguish "true" from false Christians, we can't be asked to distinguish true from false interpretations of the Bible. While I certainly hope (for instrumental reasons) that more Christians come around to the kind of pro-science, pro-social-justice strain of Christianity that Clark espouses, that's a debate that has to happen internally among Christians.
Clark's mistake comes from a thread of Christian chauvinism that runs through his otherwise admirable take on his faith. He is rightly appalled by the ends to which people have put Christianity. He believes not just that their ends are wrong, but that they are not true to Christianity, and that the religion can and should be better than that. So far, so good. But in his desire to redeem the faith, to clean his own hands and demonstrate "we're not all like that," he feels entitled to enlist non-Christians in his battle. He wants non-Christians to help him cleanse Christianity of the wrong kind of Christians, so that Christianity can be good and he can avoid being tarred by association with the bad ones. But that is not our battle. Non-Christians have no stake in the fate of Christianity. As long as everyone believes in Ohio (or evolution, or gay rights ...), then it's their own business whether they infer that Christianity must be Ohio-compatible, or whether they reject Christianity as hopelessly Ohio-denying.