Short Answer
No, the Bible does not teach that the earth is flat.
Authorial Intent
If we want to understand what the Bible teaches, we have to start by asking what the authors were trying to communicate to their original intended audiences. We can not start with our own questions and try to "see what the Bible says about it". This is something you learn in introductory-level Hermeneutics classes. So the first question we need to ask when interpreting each of these passages is: Was the author trying to advance a particular cosmology? And the answer will almost always be: No.
That is not to say that Scripture is silent on cosmology or the origin of the earth or universe, but even where it does speak to these issues, cosmology was not the the primary intent behind the passage. For instance, Genesis 1:2ff and Exodus 20:11 tell us that God "made" the heavens, the earth, the seas, and everything in them in 6 days, but notice that we are only told about a select few details which pertained directly to the preparation of the place where man would interact with God -- and even then it is all from the perspective of an observer standing on earth (e.g. there were "evenings and mornings".)
The authors of Scripture were concerned first and foremost with man's relationship with God. This is why there is almost nothing in the Bible about what God was doing before Creation, the lives & histories of angels & demons, the existence of other life, parallel universes, the makeup of the atom,... the Bible is almost exclusively about man's relationship with God.
Bias and Agenda
It is important in Bible interpretation to know your bias. For example, modern interpreters will typically approach this particular topic from one of two angles. They will either come at it from the perspective that the ancient writers were primitive, and their ignorance has now been exposed through our recent advances in science, or they will take all Scripture to be the inspired word of God and attempt to show how scientifically progressive the ancient documents were. The first group will often draw pictures (like the one Mark Edward provided in his answer) and try to show how the biblical cosmology corresponds to other primitive cosmologies. The second group will pull verses out of context to show how the biblical authors were way ahead of their time scientifically. (Example.) Though both of these approaches can be very compelling, both approaches are wrong. Our goal should be exegesis, not the advancement of our preconceived notions.
In both cases the error is usually quite clear once you examine the passages they use more carefully and ask the simple question: "Is that really what the author was trying to say?" So let's get to it, and examine the passages that lead Mark Edward and others to claim that the Bible teaches a flat earth.
Genesis 1
Genesis 1 is the bedrock of biblical cosmology and origins. Virtually every other passage in Scripture on the topic is build and based on this chapter. If you misunderstand this chapter, you will almost certainly misunderstand every other passage on the subject.
1:1) The chapter opens up with a brief but powerful theological statement. In the beginning, God created the heavens and the earth. "The heavens and the earth" is a literary device known as a merism. It means "everything". The author wants the reader to know that everything was made by God -- that's where it came from. Right there you have a sharp deviation from virtually every other ancient cosmology. We call it "creation ex nihilo" (creation out of nothing.)
From 1:1 to 1:2) Scholars are divided on the relationship between 1:1 and 1:2ff.
- Some take 1:1 as chronologically prior to 1:2. In this view, 1:1 describes "creation ex nihlo" -- and perhaps the entire process of the universe's evolutionary development, while 1:2ff describes a subsequent forming / preparing of the creation for man. The strongest exegetical support for this view is the use of a different verb in 1:1 (create) than he regularly uses in 1:2ff (made).1 This is a respectable view, and has recently been championed by Hugh Ross and John Sailhamer, among others.
1) Note that the creation of man is an exception to this pattern, leading many to conclude that this was a special creation event, unlike the origin of the animals and plants.
- Others take 1:1 as a title and introductory overview to 1:2ff. In this view 1:1 is a brief statement, and in 1:2 the author moves in for a more detailed look. (1:1 also adds information, as noted above.) This is also a respectable view. I lean toward this interpretation based on the "other bookend" in 2:1-4 and the merism in Ex 20:11.
1:2) Either way, we get our first detailed look at ancient cosmology in 1:2. As I noted in another answer, what we see in 1:2 is not empty nothingness, but land, sea, and what we might think of as air. (By "air" I simply mean that the sea had a surface which could be hovered above.) This is in sharp contrast with virtually every other ancient cosmology, including Mark Edward's artwork. The author doesn't specify that the earth was a "sphere" or "flat" -- that wasn't his point. His point was that the earth was unihabitable at first. The land was under the water and there was no light. But the Spirit was hovering over the surface of the water, indicating something special was about to happen.
1:6-8) After making light, God splits the waters so that we now have waters "above" and waters "below", with an expanse called "heaven" in between them. This refutes Mark Edward's notion that the expanse was a solid crystal. (Also, in verse 20 he says that this "expanse" is where the birds fly. Did Moses think the birds flew in solid crystal above the stars? Probably not!) So now we have upper waters, lower waters and sky in between.
1:9-10) Here we see God bring the land up for air and move the waters aside, so now we have dry land and seas.
1:14-18) This is interesting, because now God "makes" (however you take that word) the sun, moon, and stars in the sky. There are two ways to take this.
Some would say that since the stars were in the sky, they were beneath the "waters above". So in this view we have land & sea, sky above the land & sea -- with stars in it, and then waters above the starry sky. It is a rare for an exegete to take this position, but it is exegetically possible.
Another more common interpretation is that the "waters above" were meant to indicate the clouds, which are not a solid mass of water as Mark Edward assumed. From a human perspective they are "above", and there is sky between us and them. So in this view you have land & sea, sky above the land & sea -- including outer space, and "waters above" which are the clouds. I lean toward this interpretation; I think this is what Moses meant. I don't think Moses was trying to teach his readers that the sun & moon were under the clouds. Even a primitive man could look up and see that the clouds often stood between the observer and the sun, moon, and stars.
The remainder of the chapter (through 2:4 I believe) continues to show how God prepared a place for man to dwell with Him, and show that He set man over His entire creation. There's a lot of theology in this chapter, but we've covered the cosmological clues pretty well, showing that Mark Edward's interpretation is untenable, and Genesis does not match other ancient cosmologies at all. On the other hand, it doesn't gel well with the popular notions of most modern scientists either.
Psalm 74:12-17
Mark brought up this passage as evidence that the author held to primitive cosmologies, but this passage is not talking about cosmology -- or even creation.
. . . God is my king from of old, Who works deeds of deliverance . . .
. . . You divided the sea by Your strength . . .
The most natural way of taking this would be as a reference to the Exodus, not to the Creation Week. I won't comment on the entire passage, but when the Psalmist speaks of God crushing the heads of sea monsters and giving Leviathan as food for those in the wilderness, we can't just jump to the conclusion that this is about God creating the world by conquering other primordial gods as Mark Eward suggested. For further study, pick up a good commentary on the passage and seek to understand the author's intent.
Psalm 104
Mark Edward suggests that this poetic song teaches us that the earth is on foundations. First, consider some of the language used in this poetic song.
He makes the clouds His chariot;
He walks upon the wings of the wind;
The earth is satisfied with the fruit of His works.
The sun knows the place of its setting.
You open Your hand, they are satisfied with good.
You hide Your face, they are dismayed;
I don't know anybody who would take this to mean that God has a hand and a face, the sun has a brain, God's activities cause apples & oranges to pop out all over the place, the wind has wings, and God likes to ride in a chariot made of water vapors. As any good hermeneutics book will tell you, you need to understand the kind of writing you are reading before you begin interpretation; you can't just take a verse out of a poetic song, take it literally, and then use it to make the Bible look ridiculous.
The author's intent was to sing poetic songs about the greatness of God's deeds. Yes, it suggests that during Creation God commanded the waters to recede and the mountains to rise. The author got that idea, of course, from Genesis 1. This isn't a standard, primitive cosmology; it's a Genesis 1 cosmology.
Proverbs 3:19-20, 8:22-36
The same thing can be said here. In the first passage, the author simply says "the deeps were broken up"; this is just basic Genesis 1 stuff, which I covered earlier. Mark Edward claimed that since this passage says the earth was "founded" this must mean that it was sitting on a physical foundation. The obvious problem with that approach to interpretation is that words have a semantic range, and the context is our only way of zeroing in on the author's usage of a term. The word clearly means "founded" here, not "on a physical foundation in primeval waters". (Note also that this statement occurs in a passage which calls wisdom a woman. The author clearly did not mean that to be taken literally, and he didn't mean for "foundation" to be taken as brick and mortar either.) The intent here was not to teach a flat earth; it was to magnify God for His wisdom during Creation.
The second passage doesn't teach a flat earth either. Mark Edward cites this passage as support that the "firmament" of Genesis 1 was a solid object. Here's what it actually says:
“When He established the heavens, I was there,
When He inscribed a circle on the face of the deep,
When He made firm the skies above,
When the springs of the deep became fixed,
When He set for the sea its boundary
So that the water would not transgress His command,
The author is clearly speaking of God's wisdom in setting boundaries for various things. Mark Edwards, however, sees the word "firm" and interprets it to mean "a literal, solid, crystaline dome over the starry sky". Not only does his interpretation ignore the authorial intent, but it doesn't even make sense to think that Solomon would say such a thing. Does he mean to suggest that Solomon was unaware that birds fly in the sky? Or that Solomon never made it to verse 20 in his readings of Genesis 1? These proverbs do not teach a flat earth, nor do they imply a primitive cosmology. They speak of God's great wisdom during Creation.
Job 26:5-14
Job does not teach a flat earth or primitive cosmology either. Since Mark Edward leaned heavily on Job for his answer, I need to cover several things here.
First, speaking of Sheol as a place "down below" was as common in biblical times as it is today. This was not a statement about cosmology. Job 26:5-6 simply teach that God is aware even of what goes on in the realm of the dead.
In 26:7 Job expresses his awe and wonder that the stuff "above" is not sitting on anything and yet it does not fall. Likewise, the stuff "below" (earth, seas, etc.) is not suspended in some way by the things above, and yet somehow is maintained below them without falling. First, this directly refutes Mark Edward's claims about biblical cosmology. Clearly Job did not believe that heaven was resting on pillars. Second, Job wasn't making positive affirmations about cosmology; he was speaking of what was not the case.
In 26:8 Job expresses his awe about the clouds. 26:9 says the clouds cover the moon. Nothing strange here. Is 26:10 speaking of an "opening" in the primordial waters? Is it teaching a flat earth? No. Look at the text more closely. The circle is "on the surface of the waters" and and is "at the boundary of light and darkness". This is probably referring to the horizon or reflection of the sun on the ocean, as the NET Bible and others have captured well:
"He marks out the horizon on the surface of the waters as a boundary between light and darkness." --NET Bible
Mark Edward's takes it as an imaginary, circular division between light and darkness in the sky, but not even a primitive man would conclude that; simple observation of the sky would refute such a notion.
- Mark claims that verse 11 teaches that heaven sits on pillars. Read the whole verse though:
“The pillars of heaven tremble
And are amazed at His rebuke.
The only way to conclude that these are literal "pillars" would be to also conclude that these literal "pillars" had personalities and could be amazed! Mark Edward's artwork has mountains holding up the sky. It is quite clear that Job was not thinking of mountains (or stone pillars) which physically support the sky. So what is Job saying? You could either take it poetically, or you could take the "pillars" in another sense as spiritual powers, but either way this verse is not helpful in reconstructing Job's cosmology; that's simply not what he was talking about. I'll stop there for now.
Conclusion
When we read Scripture we need to respect the authorial intent. We cannot allow ourselves to bring modern assumptions to the text as so many do today. Mark Edward did in his answer; he assumed that the authors were primitive and that they taught a rather standard primitive cosmology. As we have seen, that assumption is false. The biblical cosmology and story of origins is totally different from other ancient cosmologies.
The biblical cosmology -- and in particular its origin -- is really only outlined in Genesis 1 and those texts which reference Genesis 1. This unique depiction of cosmic origins was just as much at odds with primitive cosmological creation myths as it is with the notions of modern scientists. It is just as inappropriate to classify it as "primitive" as it is to classify it as "modern".
Beyond creation, the Bible's post-creation cosmology is not at all at odds with our modern observations. Scripture teaches that there is dry land, seas with land under them, "waters above" which likely are a reference to the clouds, and a sky in which birds fly and stars were placed. Job (among others) believed that the stuff "up there" was not physically connected to the stuff "down here" by pillars or ropes or anything of that sort, but rather that it was set there by God. The biblical authors were in awe of what God had made and were careful not to claim wisdom which was beyond them. Indeed, most of the passages in Job are intended to show man's ignorance about how things work, not to detail their operation.
In summary, with regards to the question of whether the Bible teaches that the earth is flat, the answer is very clearly: No.
