Zakir Naik: Touching Quraan without Wudhu

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.

Source: islamqa.com/en/ref/10672