"... According to Razi, it is on this passage, among others, that the great Qur’an-commentator Abu Muslim al-Isfahani based his rejection of the so-called ‘doctrine of abrogation’ discussed in my note 87 on 2:106." (Asad, Message of the Qur’an [Dar Al-Andalus Limited 3 Library Ramp, Gibraltar rpt. 1993], p. 443, fn. 35; online edition)

The following Muslim cites Sura 10:64 as proof that the Quran is unchangeable:

5. Why don't Muslims adapt the Quran to the needs of the modern age?

a.
The Quran states: there is no changing the words of God. (1)   (GF Haddad, Frequent Questions About Islam And Religion; online source)

In the footnote, this is what we find the author citing:

1
Those who believe and guarded (against evil): They shall have good news in this world's life and in the hereafter; there is no changing the words of Allah; that is the mighty achievement.
Sura Yunus (10) verses 63/64   (Source)

The problem is obvious: On the one hand, Sura 6:115, 6:34, 10:64, 18:27 make it clear that NONE CAN change the words of God (which is supported by the explanation of the commentators quoted). But, on the other hand, God DOES exchange one verse for another verse (Sura 2:106, 16:101). And he does so through his messengers like Jesus [supposedly changing some rules given through Moses] and like Muhammad who gives rules different again from those of Moses and of Jesus.

Muhammad Asad’s above mentioned footnote 87 on Sura 2:106 is quite interesting:

"... The principle laid down in this passage - relating to the supersession of the Biblical dispensation by that of the Qur’an - has given rise to an erroneous interpretation by many Muslim theologians. The word ayah ('message') occurring in this context is also used to denote a ‘verse’ of the Qur’an (because every one of these verses contains a message). Taking this restricted meaning of the term ayah, some scholars conclude from the above passage that certain verses of the Qur’an have been ‘abrogated’ by God’s command before the revelation of the Qur’an was completed. Apart from the fancifulness of this assertion - WHICH CALLS TO MIND THE IMAGE OF A HUMAN AUTHOR CORRECTING, ON SECOND THOUGHT, THE PROOFS OF HIS MANUSCRIPT, deleting one passage and replacing it with another - there does not exist a single reliable Tradition to the effect that the Prophet ever declared a verse of the Qur’an to have been ‘abrogated’. At the root of the so-called ‘doctrine of abrogation’ MAY LIE THE INABILITY OF SOME EARLY COMMENTATORS TO RECONCILE ONE QUR'ANIC PASSAGE WITH ANOTHER; a difficulty which was overcome by declaring that one of the verses in question had been ‘abrogated’. This arbitrary procedure explains also why there is no unanimity whatsoever among the upholders of the ‘doctrine of abrogation’ as to which, and how many, Qur’an-verses have been affected by it; and furthermore, as to whether this alleged abrogation implies a total elimination of the verse from the context of the Qur’an, or only a cancellation of the specific ordinance or statement contained in it. In short, the ‘doctrine of abrogation’ has no basis in historical fact, and must be rejected ..." (Asad, Message of the Qur’an [Dar Al-Andalus Limited 3 Library Ramp, Gibraltar rpt. 1993], pp. 22-23, n. 87; online edition; bold and capital emphasis ours)

Asad correctly points out that abrogation is an indication of human imperfections and weakness.

Another Muslim, the Maulana Muhammad Ali of the Ahmadiyya sect, rejected the doctrine of abrogation because it violates the claim of the Quran that it is free from errors and discrepancies. Yet he readily admitted that this concept was developed because Muslims were confronted with references that conflicted with one another which they could not satisfactorily explain:

The principle on which the theory of abrogation is based is unacceptable, being contrary to the clear teachings of the Qur'an. A verse is considered to be abrogated when the two cannot be reconciled with each other; in other words, when they appear to contradict each other. But the Qur'an destroys this foundation when it declares that no part of it is at variance with another: "Will they not then meditate on the Qur'an? And if it were from any other than Allah, they would have found in it many a discrepancy" (4 : 82). It was due to lack of meditation that one verse was thought to be at variance with another; and hence it is that in almost all cases where abrogation has been upheld by one person, there has been another who, being able to reconcile the two, has repudiated the alleged abrogation. (Ali, The Religion of Islam [The Ahmadiyya Anjuman Isha'at Islam (Lahore) U.S.A., Eighth Edition 2005], p. 32; bold and italic emphasis ours)

What Ali's candid admission shows is that the Muslims who appeal to abrogation do so primarily because they are unable of reconciling the errors within the Quran. Abrogation therefore becomes the convenient way of explaining away these discrepancies.

Amazingly, among the Muslims that accept abrogation are some who openly admit that the doctrine implies that there are contradictions within the Quran! Note, for example, what this Islamic site claims are the qualifications which are needed for someone seeking to be a mujtahid, i.e. a person wanting to 'exert himself' to form an opinion in legal matters:

A mujtahid should have the knowledge of nasikh and mansukh (abrogating and abrogated), i.e., which one out of two CONTRADICTORY and opposite texts is later in revelation. This might have occurred due to change of a rule, replacement, withdrawal or omission. It is not necessary to remember all such texts. But one must enquire the text related to the concerned issue. Past scholars have done a lot of work about an-Nasikh, and have listed all such verses and hadiths. Now it is not difficult to find it out anytime. (IslamOnline.net, Conditions of a Mujtahid; source; capital and underline emphasis ours)

Contradictions in the Qur'an
Answering Islam Home Page