Praise be to Allah.
The cover of a Mushaf that is attached to it with binding or glue, and the like, comes under the same ruling as the Mushaf. So it is not permissible for anyone who does not have wudoo’ to touch it. As for the cover that is separate from the Mushaf, like the sleeve in which the Mushaf is placed to protect it, there is nothing wrong with touching it without wudoo’.
It says in al-Mawsoo‘ah al-Fiqhiyyah al-Kuwaitiyyah: The majority of fuqaha’ – Hanafis, Maalikis, Shaafa‘is and Hanbalis – are of the view that it is not allowed for one who is not in a state of purity to touch the cover of the Mushaf that is attached to it, or the margins on which there is no writing on the pages of the Mushaf, or the white spaces between the lines, and the same applies to pages that are free of any writing at all. That is because they are part of the book. Some of the Hanafis and Shaafa‘is are of the view that that is permissible. End quote.
Shaykh Ibn Baaz (may Allah have mercy on him) said: It is not permissible for the Muslim to touch the Mushaf unless he is in a state of purity and free of both major and minor impurity. The same applies to moving it from one place to another, if the one who is moving it is not in a state of purity.
But if he touches it or moves it with a barrier, such as if he picks it up in a wrapper or sleeve, and the like, there is nothing wrong with that. As for touching it directly when he is not in a state of purity, that is not permissible according to the correct view, which is the view of the majority of scholars, for the reasons mentioned above. As for reciting or reading, there is nothing wrong with him reciting it from memory even if he is in a state of minor impurity, or reading when someone else hold the Qur’an for him and prompts him; there is nothing wrong with that. " (Majmoo‘ Fataawa Ibn Baaz 10/149, 150).
And Allah knows best.