The Qur'an is a sacred text sent down by Allah and should be treated with care and respect. Does that mean we need to be in a state of purity (have wudhu) when holding or reading from the Qur'an? What is the difference between a Mus'haf and a Taf'heem?
The vast majority of scholars actually agree that you should have wudhu before you touch the mushaf...
Shaykh Waleed Basyouni explains further
Alhumdulillah was-Salatu was-Salamu ‘ala Rasoulillah amma Ba’d. This question about touching the mushaf without wudhu, is it allowed? The vast majority of the Muslim scholars agree that the person should have wudhu before touching the mushaf. Because this has been reported from several companions: Al Sheik-Al-Islam Ibn Taymyyah, Al-Nawawi - and before that- said ‘Ali ibn Abi Talib, Salman al-Farsi, ‘Abdullah ibn ‘Umar said that and they did not [oppose it], we didn't hear any one of the companions oppose them or say something different than them. So that's why the ‘Ulama said it's a matter of agreement between the companions that the person should have wudhu before touching the Mushaf.
Also, they base this argument on the hadith of the Nabi (sallahu alaihim wa salam) to 'Amr Ibn Hazm, when he sent him a letter and in it: (la umus-al-mushaf illa tahir)- "Only a taahir should touch the mushaf." Because of this evidence, you'll find the vast majority of the fuqaha agreed upon that. Hanafi, Maliki, Shafi'i, Hanbali said you should have wudhu before touching the mushaf.
With this being said, I want to also say that there are some scholars who disagree with that and they said that touching the mushaf with wudhu is only recommended, and among the tabi’een, Al-Hasan Al-Basri, Ibn Sirin, and Ibn Hazm (raheemuhullah) wrote about this a lot and he basically adopted the opinion that having wudhu is not a condition to touch the mushaf.
For me, I would highly recommend you to have wudhu before touching the mushaf, and also I want to make sure to make this distinction between the mushaf, which is the Arabic text of the Qur'an, and the Qur'an which exists in any ‘daajid’, like in your phone, in your computer screen, or in a translation of the Qur'an or Book of Tafseer, you do not need to have wudhu to touch these things, and may Allah (subhanuhu wa ta’aala) always guide us to what is right.