One of the most famous cases of "religious institutions vs scientists" was the Gallileo deal. Everybody knows the story - the Catholic Church adhering to Ptolemee's (sp?) belief that the Earth is at the center of the universe with everything else revolving around it, and Copernicus's observations and calculations indicating that the Earth and other planets revolve around the Sun.
Now, a funny thing about the wholde deal was that Ptolemee's model wasn't even part of doctrine. Unless I'm forgetting something, it isn't mentionned anywhere in the Bible that the Sun revolves around the Earth. Church authorities adhered to the model because they liked it, not because it was part of their religion.
Nowadays, there are certain conflicts surrounding the Bible. Genesis in particular comes to mind, for obvious reasons. Some simply consider it a non-issue ("Jeez, people - it's just a collection of myths. Check the evidence!"). Some assume Genesis was meant figuratively ("Well, if the scientists say so, they probably know what they're saying. So, God must not have meant it litterally."). A final faction dismisses the conclusions of most of the scientific community, giving more credibility to the Bible ("God wrote it. Are you calling God a liar?").
Now, concerning the thrid faction...Something's occured to me. Is there any logical basis at all behind it? Is biblical innerancy part of doctrine? (Not that this would make it correct, mind you) Or is it just a belief that's popular with certain fundamentalists?
The Epistles were written by Paul, who admitted that their content represented his own opinions, and not necessarilly God's. The Gospels were supposedly written by disciples of Jesus, and I don't think any of them claimed to have received divine inspiration during the writing process. That leaves the Old Testament. Does it claim at any part to have been written, or at least inspired by God - rather than just, you know, being written by humans who saw/heard stuff happen and decided to write it down?
I think it is part of Muslim doctrine that an angel dictated every word in the Coran to Mohammad. I don't recall any such claim for Judeo-Christianity...