It is recommended to have wudu always. Hadith says wudu is light and wudu on wudu is light on light. (Light means spiritual benefits that we are not able to understand them. It is like explaining what is light of sun for a blind born man who never has seen sun. So hadith says only light.)
Also this verse:
وَلَمْ يَرَ الَّذِينَ كَفَرُوا أَنَّ السَّمَاوَاتِ وَالْأَرْضَ
كَانَتَا رَتْقًا فَفَتَقْنَاهُمَا وَجَعَلْنَا مِنَ الْمَاءِ كُلَّ
شَيْءٍ حَيٍّ أَفَلَا يُؤْمِنُونَ
Have those who disbelieved not considered that the heavens and the
earth were a joined entity, and We separated them and made from water
every living thing? Then will they not believe?
In this verse Water is wudu and it means with wudu soul and heart of human become alive and can receive divine inspires and gifts.
According to this verse:
لَّا يَمَسُّهُ إِلَّا الْمُطَهَّرُونَ
None touch it except the purified. http://tanzil.net/#56:79
Having wudu is needed only for touching Quran. And if you do not touch its words it is not needed. but still it is recommended.
Who are seeking higher ranks of believers and want to their spiritual eyes become open always respect recommended rules of Islam. but who only want to escape Hell not. if you want to learn deeper meanings of Quran having wudu is recommended. Hadith says Quran has 70 levels of meaning and each level has 70 levels. One scholar said one time I could have opportunity to meet a Wali of Allah and he said for me 5000 different meaning for only one verse of Quran.