Shaykh ‘Abd al-‘Azeez ibn Baaz was asked a similar question, and he said: 

It is not permissible for a Muslim to touch the Qur’aan when he does not have wudoo, according to the majority of scholars. This is the view of the four imaams (may Allaah be pleased with them), and this was the view expressed in the fatwas of the Companions of the Prophet (peace and blessings of Allaah be upon him). A saheeh hadeeth concerning that has been narrated from ‘Amr ibn Hazm (may Allaah be pleased with him), stating that the Prophet (peace and blessings of Allaah be upon him) wrote to the people of Yemen: “No one should touch the Qur’aan except one who is taahir (pure).” This is a jayyid hadeeth which has a number of other isnaads which strengthen it.

Hence it is known that it not permissible to touch the Qur’aan except in a state of purity from both major and minor impurity. The same applies to moving it from place to place, if the person who is moving it is not taahir. But if he touches it or moves it with something in between, such as picking it up in a wrapper, then it is OK. But if he touches it directly when he is not taahir, this is not permitted according to the saheeh view of the majority of scholars, for the reasons stated above. With regard to reciting it, it is OK for him to recite it from memory when he is without wudoo’, or for him to read it if the Qur’aan is held by someone who asks him to correct or prompt him. 

But the person who is junub, i.e., in a state of major impurity, should not recite Qur’aan, because it was narrated from the Prophet (peace and blessings of Allaah be upon him) that nothing ever kept him from reciting Qur’aan except for janaabah (major impurity). Ahmad narrated with a saheeh isnaad from ‘Ali (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) came out from the toilet and recited something from the Qur’aan. He said, “This is for the one who is not junub; but the one who is junub should not do this, not even one aayah.” 

The point is that the one who is junub should not recite Qur’aan either from the Mus-haf or from memory, until he has taken a bath (ghusl). But the one who has broken his wudoo’ and is impure in the sense of minor impurity may recite Qur’aan from memory but she should not touch the Mus-haf. 

Fataawa al-Shaykh Ibn Baaz (may Allaah have mercy on him), 10/150 

With regard to the hadeeth about the purity of the believer, it was narrated that Abu Hurayrah said: “I was met by the Messenger of Allaah (peace and blessings of Allaah be upon him) and I was junub. He took my hand and I walked with him until he sat down. Then I slipped away and washed myself (ghusl), then I came to where he was sitting. He said, ‘Where were you, O Abu [Hurayrah]?’ I told him, and he said, ‘Subhaan-Allaah, O Abu [Hurayrah]!, the believer does not become impure.” (Narrated by al-Bukhaari, al-Ghusl, 276; Muslim, al-Hayd, 556).

Al-Nawawi said in his Commentary on Saheeh Muslim: this hadeeth illustrates a great principle that the Muslim is taahir whether he is alive or dead. If his purity is established, then his sweat, saliva and tears are all pure too, whether he (or she) is without wudoo’, in a state of major impurity (junub), menstruating or bleeding after childbirth. 

Once this is understood, then the meaning of his being pure (tahhir) will become clear. It means that there is nothing to prevent his body from being essentially pure even when at the same time he has broken his wudoo’, because being without wudoo’ is simply something that prevents one from praying, or doing other things for which tahaarah (purity) is a pre-condition.  

And Allaah knows best.