Will the Church go through the Tribulation?
Yes, Scripture teaches that the Church will remain on earth through the Great Tribulation and be persecuted by the Antichrist, but the Church will not experience the Day of the Lord, which is the outpouring of God's wrath, since believers are not appointed to that wrath.
The Bible draws a clear line between the Great Tribulation and the Wrath of God, and it is vital not to confuse the two. The Great Tribulation is the season of persecution and affliction that the saints endure under the reign of the Antichrist. It works patience and faith in God's people, even though many will be killed for refusing the mark of the beast. Revelation 13:7 says power was given to the beast to make war with the saints and overcome them, and Matthew 24:9 warns that believers will be delivered up to affliction and death for Christ's name. This tribulation is something the Church walks through, not something it is spared from.
The Wrath of God, by contrast, is described as the seven last plagues, poured out only after the Antichrist's reign, the mark of the beast, and the days of Great Tribulation have run their course. Revelation 15:1 says the seven angels with the seven last plagues fill up the wrath of God. Paul makes clear that the Church is not appointed to this wrath: "For God hath not appointed us to wrath, but to obtain salvation by our Lord Jesus Christ" (1 Thessalonians 5:9). So while the saints suffer tribulation at the hands of the Antichrist, they are not present for God's own vengeance on the wicked, which comes afterward.
Matthew 24 lays out the order plainly: the beginning of sorrows, then the Great Tribulation under the abomination of desolation, and only then, "immediately after the tribulation of those days," the sun and moon are darkened and the sign of the Son of man appears in heaven as He gathers His elect (Matthew 24:29-31). The rapture, the gathering together of the saints, therefore comes after the tribulation, not before it. Revelation 7:14 confirms this, describing the great multitude before the throne as "they which came out of great tribulation," having washed their robes in the blood of the Lamb; the Church is shown in heaven only after having passed through that tribulation.
Revelation 6:9-11 and 14:12-13 show that during the very days of the mark of the beast, there are still saints alive on earth, keeping the commandments of God and the faith of Jesus, some of whom will still be killed. This is why it would be inconsistent to say the whole Church is removed at the opening of Revelation chapter 4; the testimony of Jesus and the patience of the saints are still described as present through the time of Antichrist. The mystery of God, Christ in His people, is not finished until the seventh trumpet sounds, the true last trump, at which point the Church's hope is fully manifested and the age of the Church on earth comes to its close.
The scriptures
For God hath not appointed us to wrath, but to obtain salvation by our Lord Jesus Christ,
And I saw another sign in heaven, great and marvellous, seven angels having the seven last plagues; for in them is filled up the wrath of God.
And it was given unto him to make war with the saints, and to overcome them: and power was given him over all kindreds, and tongues, and nations. And all that dwell upon the earth shall worship him, whose names are not written in the book of life of the Lamb slain from the foundation of the world. If any man have an ear, let him hear. He that leadeth into captivity shall go into captivity: he that killeth with the sword must be killed with the sword. Here is the patience and the faith of the saints.
Then shall they deliver you up to be afflicted, and shall kill you: and ye shall be hated of all nations for my name’s sake. And then shall many be offended, and shall betray one another, and shall hate one another. And many false prophets shall rise, and shall deceive many. And because iniquity shall abound, the love of many shall wax cold. But he that shall endure unto the end, the same shall be saved. And this gospel of the kingdom shall be preached in all the world for a witness unto all nations; and then shall the end come.
Immediately after the tribulation of those days shall the sun be darkened, and the moon shall not give her light, and the stars shall fall from heaven, and the powers of the heavens shall be shaken: And then shall appear the sign of the Son of man in heaven: and then shall all the tribes of the earth mourn, and they shall see the Son of man coming in the clouds of heaven with power and great glory. And he shall send his angels with a great sound of a trumpet, and they shall gather together his elect from the four winds, from one end of heaven to the other.
After this I beheld, and, lo, a great multitude, which no man could number, of all nations, and kindreds, and people, and tongues, stood before the throne, and before the Lamb, clothed with white robes, and palms in their hands; And cried with a loud voice, saying, Salvation to our God which sitteth upon the throne, and unto the Lamb. And all the angels stood round about the throne, and about the elders and the four beasts, and fell before the throne on their faces, and worshipped God, Saying, Amen: Blessing, and glory, and wisdom, and thanksgiving, and honour, and power, and might, be unto our God for ever and ever. Amen. And one of the elders answered, saying unto me, What are these which are arrayed in white robes? and whence came they? And I said unto him, Sir, thou knowest. And he said to me, These are they which came out of great tribulation, and have washed their robes, and made them white in the blood of the Lamb.
And when he had opened the fifth seal, I saw under the altar the souls of them that were slain for the word of God, and for the testimony which they held: And they cried with a loud voice, saying, How long, O Lord, holy and true, dost thou not judge and avenge our blood on them that dwell on the earth? And white robes were given unto every one of them; and it was said unto them, that they should rest yet for a little season, until their fellowservants also and their brethren, that should be killed as they were, should be fulfilled.
And the smoke of their torment ascendeth up for ever and ever: and they have no rest day nor night, who worship the beast and his image, and whosoever receiveth the mark of his name. Here is the patience of the saints: here are they that keep the commandments of God, and the faith of Jesus. And I heard a voice from heaven saying unto me, Write, Blessed are the dead which die in the Lord from henceforth: Yea, saith the Spirit, that they may rest from their labours; and their works do follow them.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.