Can I touch or read the Quran (or a Quran app) without wudu?

The majority mainstream position across the madhabs is that touching the physical Arabic mus'haf (the printed Quran text itself) without wudu is not permissible, based on Quran 56:79, "none touch it except the purified," and this is one of the more firmly held classical rulings. Reading Quran aloud or from memory without physically touching the text, however, is permitted without wudu by the majority of scholars (the one exception being a person in the state of major ritual impurity, janabah, who is generally asked to avoid even reciting until they perform ghusl).

Touching the Quran through a barrier — a cloth cover, a sleeve pulled over the hand, or the outer binding/cover rather than the text pages themselves — is permitted without wudu according to the Hanafi school and several other scholars, which is a practical accommodation worth knowing. As for Quran apps on a phone or tablet: since the screen displays an electronic rendering rather than the physical mus'haf text in the traditional legal sense, a substantial number of contemporary scholars hold that touching or scrolling a Quran app without wudu is permissible — though it remains recommended, as a matter of respect and out of caution given the genuine difference of opinion, to have wudu when reading the Quran in any form when reasonably possible.

Source: Quran 56:79; general Sunni fiqh on touching the mus'haf without wudu, with the Hanafi allowance for touching via a barrier; contemporary fatwa discussion on electronic Quran text, e.g. IslamQA and comparable fatwa services.