Allah Says in the Holy Quran Chapter 56 Surah Waqiah verses 77-79
77 That this is indeed a Qur'an, most honorable
78 In a Book well-guarded
79 Which none shall touch but those who are purified.
Although the above aayah is meant to state how the Glorious Quran is honored by the noble Angels in the heavens; the scholars have opined that just as the Quran is honored in the heavens by the Angels, so should it be honored by the believers on the earth…. And thus, one, who sincerely believes in Allah and the Last Day, should touch the Glorious Quran only when one is pure and clean, and preferably when in the state of wudu.
Allah Says in the Holy Quran Chapter 5 Surah Maidah verse 6:
6 O ye who believe! When ye prepare for prayer wash your faces and your hands (and arms) to the elbows; rub your heads (with water); and (wash) your feet to the ankles.
The only acts in Islam whereby the wudu is absolutely obligatory and a pre-condition is when one is about to offer a prayer or while circumambulating the Holy Kaaba; if these two rites are performed without the wudu, it would nullify the rites.
Sunan of Abu-Dawood Hadith 229 Narrated by Ali ibn AbuTalib
Abdullah ibn Salamah said: ‘I, accompanied by other two persons, one from us and the other from Banu Asad, called upon Ali. He sent them to a certain territory (on some mission) saying: ‘You are going to a sturdy and vigorous people; hence display your power for religion.’ He then stood and entered the toilet. He then came out and called for water and took a handful of it. Then he wiped (his hands) with it and began to recite the Qur'an. They were surprised at this (action). Thereupon he said: ‘The Messenger of Allah (saws) came out from the privy and taught us the Quran and took meat with us. Nothing prevented him from (reciting) the Quran except sexual defilement.’
Sunan of Abu-Dawood Hadith 3751 Narrated by Abdullah ibn Abbas
The Messenger of Allah (saws) came out from the privy and water was presented to him. They (the people) asked: ‘Should we bring you water for ablution?’ He (saws) replied: ‘I have been commanded to perform ablution (only) when I get up for prayer.’
Sahih Muslim Hadith 725 Narrated by Abdullah ibn Abbas
The Messenger of Allah (saws) came out of the privy, and he was presented with some food, and the people reminded him about ablution, but he (saws) said: ‘Am I to say prayer that I should perform ablution?’
In light of the above guidance of the Quran and the Sunnah, although it would be preferred to touch or recite the Glorious Quran while in the state of wudu, there is no obligation and no harm if one were to touch or recite it even when not in the state of wudu or ablution. But one should strive to abstain from touching or reciting the Quran when one is in the state of ‘janaba’ or major ceremonial impurity.
There is absolutely no harm if one wishes to recite the Quran from a computer screen when not in the state of ‘wudu’; but one should abstain from reciting the Quran from any source, book, computer screen, or even from memory when one is in the state of sexual impurity (janaba).
Whatever written of Truth and benefit is only due to Allah’s Assistance and Guidance, and whatever of error is of me alone. Allah Alone Knows Best and He is the Only Source of Strength.