With the new Left Behind movie trailer now online (starring Nicholas Cage), this might be a helpful time to explain why many of us do not believe the idea of Christians be “raptured” to heaven while others are “left behind” is a biblical teaching.
Here is a summary of the arguments from John Piper:
1. The word for “meeting” the Lord in the air in 1 Thessalonians 4:17 (apantesin) is used in two other places in the New Testament: Matthew 25:6 and Acts 28:15. In both places it refers to a meeting in which people go out to meet a dignitary and then accompany him in to the place from which they came out. One of these,Matthew 25:6, is even a parable of the second coming and so a strong argument that this is the sense of the meeting here in 1 Thess. 4:17—that we rise to meet the Lord in the air and then welcome him to earth as king.
2. The wording of 2 Thessalonians 1:5-7, when read carefully, shows that Paul expects to attain rest from suffering at the same time and in the same event that he expects the unbelievers to receive punishment, namely, at the revelation of Jesus with mighty angels in flaming fire. This revelation is not the pre-tribulational rapture but the glorious second coming. Which means that Paul did not expect an event at which he and the other believers would be given rest seven years before the glorious appearing of Christ in flaming fire. Vengeance on unbelievers and rest for the persecuted church come on the same day in the same event.
3. The wording of 2 Thessalonians 2:1-2 suggests that the “assembling to meet him” is the same as “the day of the Lord” about which they are confused. But the assembling is the “rapture” and “the day of the Lord” is the glorious second coming. They appear to be one event.
Supporting this is the reference to “gathering” the elect in Matthew 24:31. Here there is a gathering (same word) but it is clearly a post-tribulational context. So there is no need to see the gathering and the day of the Lord in 2 Thessalonians as separate events.
4. If Paul were a pre-tribulationist why did he not simply say in 2 Thessalonians 2:3 that the Christians don’t need to worry that the day of the Lord is here because all the Christians are still here? Instead he talks just the way you would expect a post-tribulational person to do. He tells them that they should not think that the day of the Lord is here because the apostasy and the man of lawlessness have not appeared. . . .
5. When you read Matthew 24 or Mark 13 or Luke 21, which are Jesus’ descriptions of the end times, there is no mention of a rapture removing believers from the events of the end. A normal reading gives no impression of a departure. On the contrary, he talks as if the believing listeners and then the readers would or could experience the things he mentions. See Mt. 24:4, 9, 15, 23, 25f, 33, etc.
6. Going through tribulation, even when it is appointed by God, is not contrary to Biblical teaching. See especially 1 Peter 4:17; 2 Thessalonians 1:3-10; Hebrews 12:3-11. But even so, Revelation 9:4 suggests that the saints will be in some measure protected in the time of distress by the seal of God.
7. The commands to “watch” do not lose their meaning if the second coming is not an any-moment one. See Matt. 25:1-13 where all ten maidens are asleep when the Lord returns. Yet the lesson at the end of the parable is, “Watch!” The point is that watching is not gazing up for an any-moment-return of the Lord; it is the moral vigilance that keeps you ready at all times doing your duty—the wise maidens had full lanterns! They were watchful!
Nor does the teaching that the second coming will be unexpected lose its force if post-tribulationism is true. See Luke 12:46 where the point is that if a servant gets drunk thinking that his master is delayed and will not catch him-that very servant will be surprised and taken off guard. But as 1 Thess. 5:1-5 says, “You (believers) are not in darkness for that day to surprise you like a thief.” We still teach that great moral vigilance and watchfulness is necessary lest we be lulled asleep and fall prey to the deceits of the last days and be overtaken in the judgment.
8. The strongest pre-tribulational text, Rev. 3:10, is open to another interpretation without any twisting. It says, “Because you have kept my word of patient endurance, I will keep you from the hour of trial which is coming on the whole world, to try those who dwell upon the earth.” But to “be kept for the hour of testing” is not necessarily to be taken out of the world during this hour, and thus spared suffering. Compare Gal. 1:4 and Jesus’ prayer for his disciples in John 17:15where to “keep from” does not mean physical removal. And notice the inevitability of martyrdom in Rev. 6:9-11. The promise is to be guarded from the hour in the sense of being guarded from the demoralizing forces of that hour.
9. The second coming does not lose its moral power in post-tribulationism. New Testament moral incentive is not that we should fear being caught doing evil, but that we should so love the appearing of the Lord that we want to be pure as the Lord is pure, for whom we hope, as 1 John 3:1-3 says.
With regard to the language of being “left behind,” see Benjamin L. Merkle’s article, “Who Will Be Left Behind? Rethinking the Meaning of Matthew 24:40-41 and Luke 17:34-35,” WTJ 72 (2010): 169-79. He argues, “Although many assume that those taken in Matt 24:40-41 and Luke 17:34-35 are taken to be with Jesus and those left behind are left for judgment, this interpretation should be rejected.”
His conclusion summarizes his arguments:
 Throughout the context of these passages Jesus uses judgment language reminiscent of the Babylonian destruction of Jerusalem and the subsequent exile of its inhabitants. Those who were taken away were the ones judged by God whereas those left behind were the remnant who received grace.
 Furthermore, the teaching of Jesus confirms this thesis. In the Parable of the Weeds the Son of Man sends his angels to gather out the children of the devil and throw them in the fiery furnace whereas the wheat is left behind (Matt 13:36-43).
 The context of Matt 24 and Luke 17 also suggests Jesus is intentionally using judgment and remnant language. Such language naturally brings up images of the former destruction of Jerusalem where the enemy came and “took away” (i.e., killed) those in the city.
 Finally, the parallel with Noah and the flood in the preceding verses strongly confirms our thesis. Just as in the days of Noah the people were taken away by the great flood, so those who are not prepared will be taken away when the Son of Man returns.
You can read his arguments in more detail here.