A Muslim cannot touch mushaf without wudu. What if a non-Muslim sees someone reading Qur'an and wants to hold it? Can Muslim allow him to hold Qur'an, or is it not allowed since that person can't technically have wudu?
This is a controversial matter and there is a difference of opinion. Most scholars prohibit disbelievers from touching the Qur'an. Shaykh Ibn Jibreen said:
"The Holy Qur’aan should not be touched by anyone except those who are purified. Allaah says (interpretation of the meaning):
“Verily, the Mushrikoon (polytheists, pagans, idolaters, disbelievers in the Oneness of Allaah, and in the Message of Muhammad) are Najasun (impure)” [al-Tawbah 9:28]
On this basis, the kaafir should not be permitted to touch the Qur’aan, whether he is a Christian, a Jew, a Buddhist, a Hindu or anything else. But it is permissible for him to listen to the Qur’aan on radio or TV, or from tapes, and it is permissible for him to read the translations of the meanings of the Qur’aan which are available in different languages.
The Messenger of Allah (peace be upon him) prohibited carrying the Qur'an to the countries of enemies and justified this by saying:
so as not to fall into their hands (i.e., the hands of the enemy) [Muslim]
Therefore, this has showed that they should not be enabled to take it, but they can listen to it. Allah (Exalted be He) says:
And if anyone of the Mushrikûn (polytheists, idolaters, pagans, disbelievers in the Oneness of Allâh) seeks your protection, then grant him protection so that he may hear the Word of Allâh (the Qur’ân) [Surah At-Tawbah 9:6]
However, some other scholars adopted the opinion that touching the Noble Qur'an is permissible for non-Muslims. They said it is probable that they embrace islam by doing so. They supported their argument by mentioning the Prophet's message (peace be upon him) to Heraclius the king of the Romans in which he mentioned Allah's Saying (Glorified and Exalted be He):
Say (O Muhammad صلى الله عليه وسلم): "O people of the Scripture (Jews and Christians): Come to a word that is just between us and you . [Surah Al-Imran 3:64]
They stated that this a great Ayah from the Book of Allah and he wrote it to Heraclius. However, the correct opinion is that this is not a proof, rather it indicates the permissibility of writing an Ayah or two of the Book of Allah. As for handing over the Mus-haf; there are no authentic reports from him (peace be upon him) in this regard.
Source: Fataawa al-lajnah al-daa’imah
According to Jafari Fiqh:
If there is any insult to Quran it is Haram and should not be allowed. but it there is any hope for his guidance there is no problem.
Official website of Grand Ayatollah Sistani (Online answering by Chat)
Let's remember the event of Omar RA accepting Islam. He wanted to see what his sister and her husband was reading. But his sister insisted to clean himself. Then he took bath and recited a portion of Quran, which changed the direction of his mind. So, obviously non-Muslims can touch Quran in a clean state, like Muslims.
Did you know the Prophet (pbuh) wrote letters with the Quranic ayats to kings? Some accepted and some trampled it. Do you think that the Prophet (pbuh) didn't know what sort of kings they were?
He (pbuh) did his job of delivering the message and so should you.
Our duty is to give the message, if someone shows interest in the Qur'an please don't create hurdles for them by saying "be clean" etc.
The bath or wash will not cleanse as much as the message of Qur'an would.