Oliver is gullible because he believes things for which he has no good evidence, and he is closed-minded because he dismisses claims for which there is excellent evidence. It’s important not to fall into the trap of thinking that what counts as good evidence is a subjective matter.
What counts as good evidence for this purpose is certainly not a subjective matter, but it is a matter relative to context, including other evidence available. One of the unusual features of conspiracy-theory thinking, distinguishing it from many other kinds of bad thinking, is that conspiracy-theories tend to be unusually evidence-rich. I guarantee you that the average 9/11-Truther knows massively more real and genuine evidence about the collapse of the towers than your average person who rejects 9/11-Truth conspiracies. Very, very few people who are not inclined to believe some conspiracy are motivated to dig into the details to the extent that believers in the conspiracy are -- usually, in fact, it's only people who are irritated enough by the conspiracy theory to spend massive amounts of time and effort answering the arguments that conspiracy theorists multiply. What is more, conspiracy theorists tend to be open to changing their conspiracy theories, whereas most people who are not conspiracy theorists go around rejecting conspiracy theories out of hand without even looking at the evidence. For this reason, I think that if Oliver, accused of being closed-minded, replies that the accuser is in fact a better candidate for closed-mindedness than he is, he will often -- not always, but often -- be right. In intellectual as in moral matters, we are sometimes saved not just by virtues but also by vices on the opposite side.
What is more, most conspiracy theorists seem not to be more gullible than anyone else. Contrary to the common stereotype, most conspiracy theorists are not conspiracy theorists about everything, but only about a family of things; and I know of no evidence whatsoever that they are more likely to be duped across the board, which should be the case if gullibility were a common vice associated with conspiracy theories.
What vices might more plausibly be named as contributing to conspiracy-theory thinking? The vices have to be associated with major features that tend to structure conspiracy theories generally: suspicion of supposed expertise, hypothesizing, and emphasis on detail. Conspiracy theories tend to involve a refusal to accept the explanation that most people would accept on the testimony of someone else, and to be built by elaborate hypotheses based on minute details. Thus the intellectual vices associated with conspiracy theories will usually be the vices that oppose the ones in Cassam's Oliver example. Conspiracy theories tend to result from the vices of excessive suspicion of authority, excessive confidence in their ability to hypothesize well, and deficiency in willingness to abstract from the evidence in order to remove mere noise. All three of these seem more plausible, because they relate to typical structural features of conspiracy theories themselves.
But perhaps there might be even better candidates?