I would have spent more time reading this article in Mother Jones on the nature of persuasion and ideas, but I'm in the middle of a Facebook argument about something Rush Limbaugh said today.
Take, for instance, the question of whether Saddam Hussein possessed hidden weapons of mass destruction just before the US invasion of Iraq in 2003. When political scientists Brendan Nyhan and Jason Reifler showed subjects fake newspaper articles  (PDF) in which this was first suggested (in a 2004 quote from President Bush) and then refuted (with the findings of the Bush-commissioned Iraq Survey Group report, which found no evidence of active WMD programs in pre-invasion Iraq), they found that conservatives were more likely than before to believe the claim. (The researchers also tested how liberals responded when shown that Bush did not actually "ban" embryonic stem-cell research. Liberals weren't particularly amenable to persuasion, either, but no backfire effect was observed.)
Another study gives some inkling of what may be going through people's minds when they resist persuasion. Northwestern University sociologist Monica Prasad  and her colleagues wanted to test whether they could dislodge the notion that Saddam Hussein and Al Qaeda were secretly collaborating among those most likely to believe it—Republican partisans from highly GOP-friendly counties. So the researchers set up a study  (PDF) in which they discussed the topic with some of these Republicans in person. They would cite the findings of the 9/11 Commission, as well as a statement in which George W. Bush himself denied his administration had "said the 9/11 attacks were orchestrated between Saddam and Al Qaeda."
One study showed that not even Bush's own words could change the minds of Bush voters who believed there was an Iraq-Al Qaeda link.
As it turned out, not even Bush's own words could change the minds of these Bush voters—just 1 of the 49 partisans who originally believed the Iraq-Al Qaeda claim changed his or her mind. Far more common was resisting the correction in a variety of ways, either by coming up with counterarguments or by simply being unmovable:
Interviewer: [T]he September 11 Commission found no link between Saddam and 9/11, and this is what President Bush said. Do you have any comments on either of those?
Respondent: Well, I bet they say that the Commission didn't have any proof of it but I guess we still can have our opinions and feel that way even though they say that.