Could you please provide some examples of events or locations that are mentioned in the Bible but has proven by historians and archaeologists to be false, such as Nazareth?
Hey Jackie, I certainly can provide some example of clearly false events or locations in the Bible. As you can see, some of it will depend on what it means to "historically prove something to be false". I should also let you know that I am a little out of my area of expertise but I am interested in the answer too so I tried to research what I could. Feel free to ask me a follow-up question about any of it.
You mentioned the idea that Nazareth is “false”. To my knowledge there really was (and is) a place called Nazareth (or Nazara). The problem with Nazareth as a historical hometown of Jesus is that all of the archaeological evidence for the inhabitance of Nazareth appears to correspond to the time the Gospels were being written (or later) but not when Jesus was supposed to live. If there was a place called Nazareth at the time Jesus (was supposed to have) lived, than it would likely have been a place too small for people to recognize. So what is “false” is some part of the Bible which describes Nazareth and Jesus' relationship to it, but the place may have a historical basis later on. That's the way a lot of things in the Bible are, the story is fabricated but the writers (being from the area) typically draw upon stories from the time they are writing.
Things too big for history to miss:
There are some things that are too big for historical evidence to miss. Big occurrences which, if true, one would certainly expect to find in sources outside of the Bible and directly related religious texts. In these cases the absence of evidence is evidence of absence. It's not that historical document don't exist or the archaeological evidence is too difficult to gather, rather that every historical record tells a different story from that told in the Bible (Torah, and Quran). To me these are the Biblical events and locations in which it is easiest to demonstrate are false. A partial list of these events would include: Creation of the universe in the manner or in the order described in the Bible (Genesis 1) , a worldwide flood (Genesis 6-9) , the Tower of Babel (Genesis 11:1-9) , the existence and subsequent destruction of Sodom and Gomorrah (Genesis 13-19) , the ten plagues of Egypt (Exodus 5-12) , the parting of the red sea and mass migration (Exodus 12) , the existence of a united monarchy of Israel with the war history described in the Bible (Joshua, Judge, Samuel, Kings) , the battle of Jericho in which trumpets knock down its walls (Joshua 6:1-27) , the sun and moon stop in the sky to help Joshua keep fighting (Joshua 10:13) , the crucifixion darkness (Matthew 27:46, Mark 15:33, Luke 23:44-45) , and the mass resurrection of saints (Matthew 27:51-54) .
There are also things in the Bible not on this list which are physically impossible (for example Moses' staff turning into a snake or Jesus turning water into wine) but if they really occurred the historical record might have missed them. There are also things in the Bible which are factual errors (for example the value of pi or historical anachronisms) or contradictions, but those wouldn't necessarily mean that a false event or location is being discussed (just that whoever wrote that part of the Bible is bad at describing them). There are also things in the Bible that there is no historical evidence for or against because it might have plausibly gotten missed by the historical record (which it includes almost everything else in the Bible, for example the existence of a non-supernatural moral teacher named Jesus) - for these things you simply have to ask yourself whether you believe Bible is true until proven false (“why would they lie?”) or false until proven true (“it has so many untrue things to be automatically trusted”).
Contradictions and fabrications in the Gospels:
Most Christians feel that the historical authenticity of the Gospels is most relevant to their religion, and therefore it has been the focus of a lot of historical work (and popular writing about it). The fact that it contains four similar, often identical, often contradictory, copies of the same story also makes a discussion of its historical accuracy uniquely complicated.
Robert Price's book “The Incredible Shrinking Son of Man” is the most recommended book for a comprehensive yet accessible explanation of the problems with the Gospels. Here is a talk by David Fitzgerald who explains what is wrong with the Gospels and how they came to be written. In particular I enjoy the segment Fitzgerald takes from Dan Barker's book Godless in which the Gospels are shown to directly contradict each other so frequently that it is impossible to even to tell the story (let along think that it is true, let alone think that it is perfect). In that video Fitzgerald also hints at a fascinating idea promoted by Richard Carrier that there never was a real person named Jesus, the God-figure and theological ideas began the myth for which literal biographical details were written in later. Richard Carrier makes some of his arguments in this video. Personally I am unsure about whether I believe there was a non-supernatural person whom the story of Jesus is based on – Carrier's argument seems to be compelling but is not generally accepted by his colleagues (who typically believe that Jesus is real but not-supernatural and stories about him are highly exaggerated or fictionalized).
This youtube series explain the problem attempting to fit Jesus' life to a real timeline. Essentially the argument is that Herod, Quirinius, John the Baptist, and Pontius Pilate are all references to people who are documented but the times at which they were supposed to interact with Jesus are incompatible with each other. If one attempts to include more detail, like an astronomical event supposed to be the Star of Bethlehem, a crucifixion eclipse ], or a crucifixion earthquake than the timeline becomes even more impossible. It's abundantly clear that the birthplace of Jesus in Bethlehem (also the unlikely stories about Herod slaughtering children and Quirinius giving a census that required people to return home) was contrived to make Jesus appear to fit an old prophecy that actually wasn't that important (that is if the stories about Jesus actually do refer to a real person). To what extent the rest is fabricated seems to be unknown.
 Obviously creationism is false, but some people try to save a semi-literal interpretation of Genesis by saying that six days really means six “periods of unspecified duration”. But this is still wrong – they have planet formation and vegetation comes before the stars; birds come before land animals. Personally I think this section is better interpreted metaphorically anyway, but because people are tempted to give it a physical meaning I had to include it on the list.
 Obviously there is no geological evidence of a great flood and this is certainly something that we could expect to have. Moreover a flood occurring during the last 100,000 years would make it immensely difficult to explain the biological, cultural, and linguistic diversity that we see today. Some people claim that every culture has the same flood story, but this really isn't true. Many cultures have flood stories but they all go differently. Moreover early civilization settled in floodplains, so we have an obvious reason for there to be stories about floods of epic proportions.
 There is no archaeological evidence for an exceptionally large tower in the region. Mythological writings outside of the Bible which give figures for the height that are impossible to have been built, let alone to be build with no archaeological evidence. Linguistic studies show that languages come from each other gradually over time (not a single even “confounding the languages”), in which geographically separated regions developed distinct dialects which diverge over time and in which multiple languages fused together .
 The closest real thing to Sodom and Gomorrah are Numeira and Bab Edha Dura (respectively). Two places within the vague region that Sodom and Gomorrah were supposed to be, but there is no evidence that these places were ever referred to as Sodom and Gomorrah, were large enough to be considered cities, or were occupied at the same time the Biblical story takes place. Both places appear to have been destroyed (in separate events) before the story was written, and therefore may have been the inspiration for the story. Numeria may have been destroyed by erosion and/or an earthquake; Bab Edha Dura may have been destroyed by an asteroid impact or a natural gas explosion.
 I wrote an AllExperts post which begins by attacking the biggest source of misinformation on this, Exodus Decoded http://en.allexperts.com/q/Atheism-2724/2012/10/atheism-14.htm
 Parting the red sea you would be a trick that you would think would be noteworthy enough for an Egyptian to reccord if it had actually happened. Moreover, the Bible describes at least 600,000 people leaving in the Exodus which is too great a number not to leave a massive trail of evidence. Moreover it would have been 15-20% of the population of Egypt at the time, so even if there was a mass exodus of people who left by non-supernatural means in a relatively short amount of time you would still see the effects in Egyptian history. There may have been Jews in Egypt at the time, but Exodus has to be mostly fiction.
 To quote Thomas L Thomson: “There is no evidence of a United Monarchy, no evidence of a capital in Jerusalem or of any coherent, unified political force that dominated western Palestine, let alone an empire of the size the legends describe. We do not have evidence for the existence of kings named Saul, David or Solomon; nor do we have evidence for any temple at Jerusalem in this early period. What we do know of Israel and Judah of the tenth century does not allow us to interpret this lack of evidence as a gap in our knowledge and information about the past, a result merely of the accidental nature of archeology. There is neither room nor context, no artifact or archive that points to such historical realities in Palestine's tenth century. One cannot speak historically of a state without a population. Nor can one speak of a capital without a town. Stories are not enough.”
 Jericho is a real place, but it had been deserted about a century before the battle of Jericho was supposed to have taken place. So it wasn't trumpets that brought down its walls, although doubtlessly the Israelites were happy to take credit for it.
 Obviously stopping the sun in the sky is not a local event and doing it for a whole day could not escape notice. No other culture has a record of this happening.
 The most common interpretation for the three hour darkness described is a solar eclipse. A total solar eclipse only lasts 7 minutes, partial eclipse “darkness” lasts about an hour, and anything less than full daylight could last up to two hours. Three hours, however, is physically impossible for a single location to endure due to an eclipse. There were about four solar eclipses in the region between 27 AD and 40 AD which may have served as inspiration for the story but did not pass through Jerusalem. It should be said that for a solar eclipse to be “caused” (ie to suddenly move into place when it otherwise would not have been on trajectory to) would require a calamitous astronomical event that would neither be physically possible nor able to avoid notice (were it to occur).
 Even the Bible doesn't refer back to this after mentioning it.
 The standard way to look for verification of supernatural events described in the Bible is to go to the historical evidence. If something breaks the laws of physics but lots of different people saw it and documented it than someone might be supposed to have really happened. But if witnesses documented that nothing supernatural happens, than someone might suppose it didn't really happen. In other words, people assume that the laws of physics can be temporarily suspended for a miracle, but the “laws of history” cannot be. As a physicist, that seems silly to me. A universe in which the laws of physics can be suspended makes is a universe that make no more sense than one where the laws of history can be suspended. It is only our personal familiarity with historical evidence (over physics) and our love of supernatural stories that makes people inclined to think this way, but I don't see any logic in it. In other words I think we should have already been able to categorize the Bible as a book of fictional stories even before we learn the historical references are wrong or flawed.
 1 Kings 7:23 describes an object with diameter 10 cubits and circumference 30 cubits. That gives us a value of pi = 3, which is mathematically wrong. If the passage were to round to the nearest cubit than it'd give a circumference of 31 cubits, so it would appear the imprecision is unnecessary.