Praise be to Allah.
Distributing leaflets and pamphlets for da‘wah purposes is a good deed and is calling people to Allah. It is something from which many people benefit, especially non-Muslims. It is a means of spreading the religion of Allah, establishing proof for them and leaving no excuse for them, and conveying to them the message of their Lord. But this work should be done in a thoughtful manner, following guidelines and methods through which the aim may be achieved in an appropriate manner without going against the rulings of Islam.
Leaflets and pamphlets for da‘wah purposes that contain Qur’anic verses do not come under the same rulings as the Mushaf, and the one who touches them cannot be said to have touched the Mushaf, because the Qur’anic material in them is mixed with other material. So they come under the same ruling as the ruling on books of fiqh, tafseer and the like, and it is permissible for a non-believer or one who is in a state of impurity to touch them.
Ibn Qudaamah (may Allah have mercy on him) said:
With regard to the verse that the Prophet (blessings and peace of Allah be upon him) wrote – i.e., to Heraclius – his intention thereby was to correspond, and a verse in a letter or a book of fiqh and the like does not come under the prohibition on touching it, and the book or letter does not become a Mushaf.
End quote from al-Mughni, 1/109
Shaykh Ibn Baaz (may Allah have mercy on him) said:
Some of the scholars are of the view that it is permissible for a non-believer to touch the Mushaf if there is the hope that he will become a Muslim. They quoted as evidence for that the fact that the Prophet (blessings and peace of Allah be upon him) wrote to Heraclius, the ruler of Byzantium, (a letter containing) the verse in which Allah, may He be glorified and exalted, says (interpretation of the meaning): “Say (O Muhammad SAW): ‘O people of the Scripture (Jews and Christians): Come to a word that is just between us and you” [Aal ‘Imraan 3:64]. They said: This verse is a verse of the Book of Allah that he wrote to Heraclius. But the correct view is that this is not proof; rather it only indicates that it is permissible to write one or two verses of the Book of Allah. As for handing over the entire Mushaf (to a non-Muslim), it is not proven that the Prophet (blessings and peace of Allah be upon him) did that.
End quote from Majmoo‘ Fataawa Ibn Baaz, 24/340-341
If these books or leaflets are translated into another language, other than Arabic, then the matter is easier. It does not matter if the non-believer touches a translation of the meanings of the Holy Qur’an into a language other than Arabic, even if it is a translation of an entire soorah, because the translation is an interpretation of the meanings of the Qur’an, and it does not come under the same rulings as the Qur’an. The non-believer is not forbidden to touch books of tafseer or Islamic knowledge, unless he is doing so by way of mishandling them or showing disrespect. See the answer to question no. 119323.
To sum up:
There is nothing wrong with distributing these leaflets and pamphlets for da‘wah purposes, even if they contain some verses of the Holy Qur’an or hadeeths of the Prophet (blessings and peace of Allah be upon him), and there is nothing wrong with allowing the non-believer to touch them and read them, because of the great interests that are served by that, of acquainting him with the religion of Allah, and establishing proof for him and leaving him with no excuse by conveying the message. The interests of conveying the message to him in the hope that he might become Muslim take precedence over the possible negative consequences of allowing him to touch them and handle them. Disallowing the printing of these books and leaflets that are aimed at non-Muslims would cancel out a great deal of good and impose restrictions on the means of calling non-Muslims to Allah.
With regard to throwing these leaflets into the trash, it is not permissible according to Islam, but it is only haraam for the one who does that action himself. As for the one who prints and distributes these leaflets and spends on that, he will have (the reward of) his good deeds and is not responsible for the actions of others.
See the answer to question no. 39376
If it is possible to write on them a request not to throw them on the ground or the like, or asking the reader to leave them in an appropriate place, that would be a good idea.
And Allah knows best.