What is Tahrif? Do the Shīʿa believe in the distortion of the Quran?

The Orthodox Shi'i View is that the Qur'an has not been Tampered With

Sheikh Saduq, one of our earliest scholars and the author of multiple books of Shia hadith writes:

Says the Shaykh Abu Ja'far: Our belief is that the Qur'an, which Allah revealed to His Prophet Muhammad, is (the same as) the one between the two boards (daffattyn). And it is that which is in the hands of the people, and is not greater in extent than that. (Al I'tiqadat)

Sayyed Al Khoei also explains likewise that the predominant view among both the classical scholars and modern scholars is that the Quran is free from alteration:

The accepted view among Muslims is that no alteration has occurred in the Qur'an, and that the text that is in our hands is the whole Qur'an that was revealed to the great Prophet (peace be upon him and his progeny). A large number of prominent scholars have proclaimed this. Among them is the leading traditionist (muhaddith) Muhammad b. Babawayh. He has counted the belief in nonalteration of the Qur'an among the doctrines of the Imamite (Twelver) Shi’ites. The jurist-doctor of the Imamite Shi’ite community, Abu Ja'far Muhammad b. al-Hasan al-Tusi, is another major figure who holds this view. He puts forth this view at the beginning of his exegesis of the Qur'an, entitled al-Tibyan, and has also cited the opinion, to that effect, of his teacher, al-Sharif al-Murtada, supporting it with the most complete evidence. (Bayan fi Tafsir al Quran)

And these views are even supported by reliable Shia hadith:

"And from their rejection of the Book was that they established its letters and distorted its Limits (I.e. narrated it more or less correctly, but didn't act on it correctly and distorted the meaning). So they were narrating it but were not acting upon it." (Al Kafi)

So there should be no doubt that the accepted view among the Shia and our earliest scholars is that the Quran is free from distortion. 

What is Tahrif?

Sayyed Khoei lists out a number of meanings of tahrif, and goes over which are problematic and which are not in his Bayan fi Tafsir al Qur'an, his book where he refutes tahrif and establishes the authenticity of the Qur'an:

The meaning of tahrif then, is about whether we can have trust that the book in front of us contains God's words, and whether we can rely on its meanings and injunctions, and there is no doubt that this is the case.

For more details, see here "Tahrif, Distortion of the Quran, A Shi'i Perspective."

Believing in the Incompleteness of the Mushaf is Not Kufr

Even though the majority of Shia scholars believe in the completeness of the Quran and its freedom from defects and errors in its general preservation, Shia scholars do not excommunicate those who believe that the current copy of the Mushaf that we have in our hands is incomplete and not a true reflection of the total size of the real Quran. This is because believing that the Mushaf is not a complete copy of the Quran does not involve denying any part of the revelation or claiming that what we have in our hands is not part of divine revelation. Some may object by saying that the Quran says: "Verily, We Ourself have sent down the Reminder (the Qur’an) and verily We, (Ourself) will be its Guardian" (15:9). However, all Shias believe that the Imam is alive and currently preserves the Quran from all corruption, so it is being preserved and therefore the verse is not being violated. 

For more details, see: Why Do Shi'as Not Excommunicate Shi'a Scholars Who Believe in Tahreef?

 How Do We Deal With Hadiths Alluding to Tahrif?

First of all, what is clear from the hadiths is that the Imams endorsed the copy of the Quran that has reached us. This means that the copy of the Quran in our hands is a means for guidance and is therefore free from distortion in any real sense. This can be proven from the following:

As for the possibility that alteration was intentionally introduced by the two caliphs in those verses that were inimical to their leadership, that too should be ruled out completely. Indeed, the Commander of the Faithful ['Ali] and his wife, the veracious and pure [Fatima] (peace be upon both of them), and a group of their supporters, had opposed the accession of Abu Bakr and 'Umar to the caliphate. They contended against them on the basis of things they had heard from the Prophet, and they called on those of the Helpers (Ansar) and Emigrants (Muhajirun) who had witnessed these events to teslify to their authenticity. They also contended against Abu Bakr by means of the Ghadir event [in which the Prophet had nominated 'Ali as his successor] and other such traditions. Al-Tabarsi mentions, in his book al-Ihtijaj, that twelve persons argued against the succession of Abu Bakr and produced textual evidence to support their argument against him. In addition, the well-known scholar al-Majlisi compiled a chapter on the subject of the Commander of the Faithful's ['Ali's] vindication of his rights in the matter of the caliphate. Had there been something in the Qur'an inimical to their leadership, it would certainly have been more worthy of mention in these arguments, and more deserving of calling upon all Muslims to witness, especially since the issue of the caliphate according to those [who believe in the alteration of the Qur'an] became an issue much earlier than the date of the Qur'an's collection. The fact that the Companions did not mention anything [about the alteration], neither at the beginning of the caliphate nor after the caliphate had fallen to 'Ali, is the irrefutable proof that the said alteration [under the first two caliphs] did not occur. (Bayan fi Tafsir al Qur'an)

Second, all of the hadiths alluding to tahrif are either (1) are weak in chain, or (2) can be understood in a way that does not involve tahrif.

Ayatollah Ma'rashi Najafi: The chief proofs for taḥrīf are the narrations and they are of different categories. More than half of them are weak in their chains, transmitted from the ghulāt, the extremists in their love [for the Ahl al-Bayt]. They forged them to relieve the rage in their hearts when they saw the Imāms (upon them be peace) impeded from their status, so they made up these narrations 

Allamah Balaghi: The vast portion of the narrations [on distortion] have chains that go back to a few individuals who have been described by the scholars of rijāl as either: weak in ḥadīth, of corrupt doctrine (fāsid al-madhhab), shunned in narration; or as inconsistent (muḍṭarib) in his ḥadīth and madhhab, that his ḥadīth is [sometimes] recognised and [sometimes] denied, and that he narrates from weak narrators; or as an accused liar from whose tafsīr it is not lawful to narrate a single ḥadīth, and that he is well-known for being a Wāqifī and the most severe of people in hatred for [Imām] al-Riḍā (upon him be peace); or that he was an extremist (ghālī) and a liar; or that he is weak, not turned to, cannot be relied upon and from the liars; or that he is corrupt in narration and accused of ghulūww. 

Ayatollah Burujerdi: As for the reports about [the Qur’ān’s] alteration, although though they are numerous from both schools [Sunnī and Shīʿa], it appears to the one investigating that the majority of them—close to two-thirds—are narrated from the book of Aḥmad b. Muḥammad al-Sayyārī, one of the secretaries of the Ṭāhirids. The weakness of his madhhab and the corruption of his creed is known by those who are well-informed about the conditions of the narrators. And many of the reports—close to one-quarter—are narrated from Tafsīr Furāt b. Ibrāhīm al-Kūfī and, like al-Sayyārī, he is of corrupt doctrine. Although, most of the reports omit the intermediary or are obscure. And many of them are clearly forgeries, such as what was reported about the name of ʿAlī (upon him be peace) being explicitly stated in the verse of tablīgh (5:67) etc. For if it was explicitly mentioned, ʿAlī (upon him be peace) would have used it as an argument in his objection with others regarding Imāmah. Another example is what was reported about the verse “If only I were dust” (78:46) originally being “If only I were a Turābī” and other [such reports] which are known to be lies.

All of the above quotes were taken from "Reflections on Mutawatir Ma'nawi"

For example, many bring the following report as evidence that Shias believe in tahrif: Muhammad ibn Yahya has narrated from Ahmad ibn Muhammad from ibn Mahbub from ‘Amr ibn abu al-Miqdam from Jabir who has said that he heard abu Ja‘far (a.s) say the following. “No one from the people has claimed to have collected the whole of the Holy Quran (in a book form) as it was revealed. If anyone would come up with such a claim, he is liar. No one collected this Holy Book and memorized as Allah, the Most Holy, the Most High revealed it except Ali ibn abu Talib (a.s) and the Imams after him.” (Al Kafi). 

However, we can easily interpret this hadith as meaning that only the Imams know the proper interpretation and explanations of the Quran, and no one can claim to know otherwise. And this interpretation is supported by another hadith in the same chapter: Muhammad ibn al-Husayn has narrated from Muhammad ibn al-Hassan from Muhammad ibn Sinan from ‘Ammar ibn Marwan from al-Munakhkhal from Jabir from abu Ja‘far (a.s) who has said the following. “No one is able to claim that with him is the whole of the Holy Quran; its apparent and hidden essence, except the executors of the will of the Holy Prophet (s.a.). (Al Kafi)

It is narrated to us by Muhammad Bin Al Husayn, from Muhammad Bin Ismail, from Mansour Bin Yunus, from Ibn Uzina, from Fuzeyl Bin Yasaar who said, ‘I asked Abu Ja’far about this report, ‘There is nothing from the Quran, except and for it is an apparent, and a hidden (meaning)’. He said: ‘Its apparent is its Revelation, and its esoteric is its explanation. From it is what has already passed, and from it is what has not yet happened, flowing just as the sun and the moon flow, just as the explanation of something from it can happen to be upon the dead, just as it can happen to be upon the living. Allah Says: And none know its interpretation except Allah, and those who are firmly rooted in the Knowledge. [3:7], we know it’’. (Basa'ir al Darajat)

Another example are hadiths which seem to add words in the verse. For example: Ali Bin Ibrahim, from his father, from Ibn Abi ‘Umayr, from Uman Bin Azina, from Bureyd Bin Muawiya who said, ‘Abu Ja’far (a.s) recited “[4:59] O you who believe! Obey Allah and obey the Messenger and those in authority from among you; then if you quarrel about anything, refer it to Allah and the Messenger, and to the Guardian (a.s) of the Command among you (Ul-Al-Amr)”. Then said: ‘How can He (azwj) Command for their (a.s) obedience and then allow them (the people) to dispute with them (a.s). But rather, He (azwj) has Said that to the Commanded ones when He (azwj) Said to them (the people): “Obey Allah and obey the Messenger”. (Al Kafi). 

Here, the Imam adds at the phrase "and to Ulil Amr among you" at the end of 4:59. At first glance, this would seem like it is indicating that the verse was distorted and the Imam is trying to recite how he believed it was revealed. However, this was a commonplace way of explaining verses, even in Sunni tafsir, and there are many examples of this same type of tafsir from Sahaba like Ibn Abbas. 

And even where the Imam says "this is how it was revealed," we refer back to the earlier hadith from Basair al Darajat which explains that the revelation also includes the explanation and interpretation of the verses as well. So there is no problem with the Imam saying such phrases when he recites a verse differently. All of the above regarding alternative explanations of hadiths which allude to tahrif is explained in greater detail in "Tahrif (Distortion) of the Qur'an a Shi'i Perspective." 

Conclusion

The mainstream Shia belief is that the Quran is not distorted. Countless hadiths from the Imams endorse the copy of the Quran in our hands, and all hadiths alluding to tahrif are either weak in chain or can be interpreted as divinely revealed explanations of the verses.