Yes. One will be sinful to touch the Holy Qur'an without Wudu which is the view of the majority of scholars. If however the verses of the Qur'an are part of a larger Tafsir or commentary then though it is still praiseworthy and respectful to have Wudu, it will not be obligatory.
As for reciting the Holy Qur'an, without touching it, it is allowed. However for a person in a major ritual impurity requiring ghusl, for example, a woman during her period or a person who has had a sexual emission during the night, it is also forbidden to recite the Holy Qur'an even from memory.