The Arabic word referring to this corruption is "tahriif", and the teaching is that the Christian (and thus Jewish) Scriptures are profoundly flawed and are thus not reliable indications of God's will, commandments, prophets, or of the history of his people. (Note: I am particularly occupied with questions regarding Jesus' teaching--the injiil--in this article.)
Tahriif is a very wide-spread belief among Muslims today, though it is not universal. It places Muslims in a very powerful position regarding the Bible because for anything they agree with already they can say, "This is from Allah, do you see that we believe in the Prophets? Why do you not accept Allah's final prophet (Muhammad) as well?" But if they are confronted with anything that challenges Islam or the Qur'an, they can say, "Clearly the text has been corrupted, for Jesus would never have said such a thing and this goes against the Qur'an."
So how can a Christian react to this? There are several paths that come to mind, all of which have been used with some degree of success in the past, I will outline three of them for you and indicate which one I generally prefer:
a) Historical Weakness of Tahriif: The traditional doctrine of tahriif tells the following story: Prophets like Jesus and Moses received genuine verbatim revelations from God like Muhammad. But perfidious Jews and Christians later corrupted these texts to suit their purposes. Now this story is found nowhere in the Qur'an, which most Muslims don't know, but it is based on certain hadiith and most Muslims simply don't know the Qur'an very well. This story is problematic from an historical point of view and prompts raises more questions than it answers:
• The New Testament took form on three different continents (Asia, Africa, and Europe), how were Christians able to collect all the valid versions of the injiil (the book Jesus received from God) given that it was recorded and spread over such a broad area?
• Languages: Sections of the NT had been translated from Greek into Latin, Syriac , Hebrew, and probably Amharic and Coptic by the end of the 2nd Century. Did the tahriif of the injiil take place before or after the translations? If before, then it must have been in the 1st Century, but there were still people alive then who had known and listened to Jesus personally, which means that there would be some record of people objecting to this corruption of the injiil. If the tahriif occurred after the translations, then we would have to believe that EVERY COPY of the true injiil which was present in three continents in (at least) half a dozen languages was destroyed.
b) Motive of Corrupters: The Gospels as we have them today give a rather unflattering picture of the Apostles. They are often foolish, proud, and faithless. It is thus unlikely that they would have "corrupted" the injiil without revising the many events which make them look foolish and weak.
It is also unlikely that one would corrupt a text without removing all the promises of persecution and familial strife, and the advocacy of poverty--hardly the kinds of things you put in a religious message if you want it to be popular. Rather, one thinks that a religious message crafted to gain popularity would include the promise of riches, women, and power--promises we do find explicitly in the Qur'an.
Moreover all the Apostles minus John were martyred for their faith. If they had corrupted the injiil and knew it to be a false message, it is inconceivable that they would all die rather than denounce it's validity. Who then corrupted the injiil? People will not die for a message they know to be corrupted. By the end of the Apostolic era the texts of the four Gospels were too wide spread and and existed in too many languages for a unified and viable tahriif to be possible.
But historical reasoning and evidence will generally get you nowhere at all with most Muslims because in Islamic culture the entire discipline of history has largely been subsumed as a sort of devotional exercise to prove the Qur'an is correct. Trying to argue from historical evidence that the Qur'an is not true is like me trying to convince you from the hymnal that the Bible is false.
c) Corruption in your Heart: This is by far my preferred apologetic. I like this approach because in many ways it mirrors Jesus' Sermon on the Mount and the interiority of the Kingdom of God. Fasting is not about letting other people know you are fasting, it is about God knowing you are fasting, for example. Similarly, I start by explaining that tahriif is very real and it is a significant problem. We say that God knows the heart of all men, and that he is all powerful--no Muslim will disagree. Then I explain that when the Jew knows the Torah and its commands, and he disobeys it, he has corrupted the Torah in his heart. Likewise the Christian who knows the commands of the injiil and disobeys it has corrupted the injiil. And finally, the Muslim who knows the commands of the Qur'an and disobeys it, is it not true that he has corrupted the Qur'an? The answer, in my experience, is always yes. Muslims are very aware that most Muslims aren't very strict in their obedience.
In conclusion I usually ask, "Is Allah powerful or weak?" Powerful! "Is Allah wise or foolish?" Wise, the answer comes. "Yes my friend, and Allah is more powerful than the Jews and the Christians, and no one is capable of corrupting God's words to his prophets! If anyone says that his words corrupted IN THE TEXT of the Torah and Gospel, he is a man who believes that God is neither wise nor powerful. But you see that corruption is in our hearts."
This is not simply a play on words. It does accommodate the Muslim who already believes in tahriif. All you have done is to reformulate the doctrine of tahriif in a way that is very much based on Jesus' teaching. But this apologetic also gets to a very foundational weakness in the Islamic view of God. God is always connected with power in Islam--Allahu Akbar! God is the greatest. Yet the Islamic narrative proposes that Jesus was not crucified, for God would not let than happen to a prophet of his. Yet we are also told that the word God gave to this prophet was not preserved? To preserve the true injiil would have been easy for God. Why did he not do it? Why did he allow 600 years of humanity to operate under the assumption that this corrupted injiil was in fact valid and accurate? And given that even from the beginning Christianity was riven with heresies and fanatics, how is that not even one copy of the real injiil was preserved by a dissenter.
All good questions. But they are not to be used to as a weapon to assault a Muslim. Whether the tone is light and conversational, or adversarial--which is sometimes necessary--we must always speak the truth in love. It is a sign of God's love to us that his true Word is in fact his Son who came to live with us, hunger with us, eat with us, cry with us, and suffer for us. His Word is not some book that one can close and place a shelf, but one who is alive and whom death could not hold down. Because of his life we have hope for eternal life: "And this is everlasting life, to know you the one true God and Jesus Christ whom you have sent."