A surprising fact about flat Earth beliefs is that they are not as old as you might have been told. A cultural myth is that man has believed the Earth is flat ever since we evolved from primates all the way up until the 17th century. And of course, it is claimed that even the Bible is guilty of this blatantly false view, proving beyond all doubt that it is not the infallible word of the Living God.
In reality, flat earth views were particularly rare even in the Biblical era. Some Greek philosophers were even able to calculate the circumference of the Earth almost correctly (they were off by a few hundred miles). According to Stephen Jay Gould, known in many Christian circles as history's most honest Evolutionist, "...there never was a period of 'flat Earth darkness' among scholars (regardless of how the public at large may have conceptualized our planet both then and now). Greek knowledge of sphericity never faded, and all major medieval scholars accepted the Earth's roundness as an established fact of cosmology." (1). The myth that everyone used to believe in a flat Earth was actually propagated by Washington Irving in the 19th century when he claimed Christopher Columbus was told he could sail far enough that he would fall off the edge of the earth. There is no evidence, however, that Columbus ever encountered such an opposition.
But while we know flat earth beliefs were almost as non-existent in history as they are today (ironically, maybe even more so), the fact remains that people did exist in the past who believed the Earth was flat. Many unbelievers misquote the Bible to say that its authors were such people. Matthew 4:8 is a favorite target for them, as it claims Satan took Jesus up a mountain and showed Him "all the kingdoms of the world". To the uninformed reader, it does sound like the Bible really is saying the Earth is flat. History buffs, or even modern linguists, by contrast, won't be even remotely phased by such an argument.
In English, "world" is a homonym (a word with identical or similar spelling/pronunciation with multiple meanings). It can mean the spherical planet on which we live, but it can also refer to a portion of the world (e.g. "third world country"), or even an intangible concept (e.g. "the world of insects"). In Greek and Hebrew, the same applies. The Bible does talk about the world as a whole, but it also refers to localities. For example, in Luke 2:1, Caesar Augustus declares that all the world must be registered. However, Caesar Augustus only really ruled Imperial Rome, so neither his decree, nor the verse recording it, were using the word "world" in a global context.
Clearly, Jesus seeing all the kingdoms of the world from a high mountain does not necessarily require a flat Earth. To be as generous as possible to the skeptic, we'll grant that if the Bible wasn't true, it could be a possible interpretation. But even in such a scenario, there are other possible, and more likely interpretations not involving a flat earth. The most likely, and I would of course say the correct interpretation, is that Satan showed Jesus all the kingdoms of the region.
Trying to force a flat earth interpretation is, at best, a very weak argument. It is clutching at straws. It is this kind of attack that makes me feel especially comfortable as a Christian, because if a popular way to "beat" Christianity is to effectively say that the Bible might say the Earth is flat, but only if you insist on taking a very particular interpretation that is in no way the only possible interpretation, Christianity must be a pretty strong faith. Truly false ideas can be refuted with far stronger arguments, and of course the most logical thing to do in any discussion is to present your strongest arguments first. Thus, if unbelievers present this view so often, it can be assumed they have little else, and thus Christianity must be airtight.
References
1. Gould, Stephen .J. - The Late Birth of a Flat Earth, 1997