No one may touch the Qur'an except one who is in a state of ritual purity.

Question:
What is the ruling on touching the mushaf without wudu, or moving it from one place to another, given that the person touching it is physically pure?

Answer:
Touching the mushaf without wudu' is not permissible according to the majority of scholars, and this is the position of the four imams (may Allah have mercy on them). It is also what the Companions of the Prophet (peace and blessings be upon him) used to issue rulings on, namely that one does not touch the Qur'an except while in a state of purity. There is a sound hadith in this matter, with no objection, from the narration of 'Amr ibn Hazm that the Prophet (peace and blessings be upon him): . It is a good hadith with various chains that support one another.
This is the obligatory ruling. Likewise, moving the mushaf or shifting it from one place to another is not to be done except by one who is in a state of purity, or if it is done by means of an intermediary, such as taking it in a wrapping, or if the mushaf is in a wrapping and he takes it by the wrapping. As for taking it directly with his hands while not in a state of purity, it is not permissible according to the correct view held by the majority of scholars.
As for recitation, there is no harm in reciting from memory while in a state of minor ritual impurity, or in reciting while someone holds the Qur'an for him and opens it for him. There is no harm in that.
However, one in a state of major ritual impurity (janabah) does not recite; because it is established from the Prophet (peace and blessings be upon him) that nothing prevents him from recitation except janabah. Ahmad narrated with a good chain from 'A'ishah (may Allah be pleased with her) that the Prophet (peace and blessings be upon him) came out from relieving himself and recited some of the Qur'an, and he said: .
The point is: one who is in a state of janabah does not recite from the mushaf or from memory until he takes a full ritual bath (ghusl). As for one who is in a state of minor ritual impurity and not in a state of janabah, he may recite from memory, but he does not touch the mushaf.
Here is an issue related to this, which is the menstruating woman and the woman with post-natal bleeding: do they recite or not? There is scholarly disagreement on this: some said they do not recite. And some said: they may recite from memory without touching the mushaf; because their duration is long—that is, the duration of menstruation and post-natal bleeding—and it is not like the state of janabah, where one takes a bath immediately and recites. But the period of menstruation may be long, reaching ten days or so, and the period for the woman with post-natal bleeding is also longer.
So the correct view is: there is no barrier to their reciting from memory, and this is the stronger position. It is established in the two Sahihs from the Prophet (peace and blessings be upon him) that he said to 'A'ishah when she menstruated during Hajj: .
The pilgrim recites the Qur'an, and the Prophet (peace and blessings be upon him) did not exclude her from this; so that indicates the permissibility of recitation for her. Likewise, he said to Asma' bint 'Umayyad when she gave birth to Muhammad ibn Abi Bakr at the miqat during the Farewell Pilgrimage; this indicates that she recites, but without touching the mushaf.
As for the hadith of Ibn 'Umar from the Prophet (peace and blessings be upon him) that he said: , it is a weak hadith. In its chain is Musa ibn 'Utbah, and in its chain is Ibn 'Iyash from Musa ibn 'Utbah. The scholars of hadith weaken the narration of Musa, and they say: he is good in his narration from the people of his homeland, the people of Sham, but he is weak in his narration from the people of Hijaz. And this hadith is from his narration from the people of Hijaz; so it is weak.