This is a popular misconception that is often used by opponents to creation to discredit and separate any view that might have a religious origin from those of naturalism. In science it should make no difference what the origin of an idea is so long as it can be tested using the methods of science. Unfortunately religion has been used as a pretext for prejudicially discrediting ideas that are testable. This is wrong.
With respect to Galileo and Copernicus, the popular mythology of a church suppressing their views is mostly false. First, Copernicus's scientific opponents did not assert that the Earth was flat, on Biblical or other grounds. They, like almost every single educated person in the west for a millenium or more, held the Earth to be round in accord with Aristotle's and Ptolemy's doctrines. Historian Jeffrey Russell has written an insightful account of events surrounding the controversy in his book, Inventing the Flat Earth. This scholarly but readable account makes it clear that the popular assertion that medievals believed in a flat earth was a 19th century invention. The assertion that anyone was burned at the stake for advocating a round Earth is also groundless. There is also no strong basis for the myth that scientists were burned at the stake for supporting Copernicus' view that the earth rotates. The only possible candidate for such a fate was one Giordano Bruno, who at the same time advocated much more serious heresies that were entirely religious in nature. Most likely he was burned for these, not for his support of Copernicus, although it is difficult to know for sure, because the records of the proceedings were burned during Napoleon's invasion.
2010 Arthur V. Chadwick, Ph.D.