After recitation, we should collect the copies of the Qur’an in a respectful manner, and put them in the proper place. This is one of the highly recommended and liked etiquettes among the Muslims around the world, and they would frown upon leaving this Divine Book of such a high status, lying around after recitation.
Imam Ja’far al-Sadiq (a) is reported to have said that 
whoever honors the Holy Qur’an will be honored on the Day of Judgement. (al-Kafi Vol. 2 Pg. 602, H. 14). One of the best ways to honor the Qur’an is to put it in an appropriate place. 
In order to appreciate the high status of the Holy Qur’an, let us look at another hadith from the Holy Prophet (saww). He (s) said, “One who recites the Qur’an and thinks someone else has been granted better than him has looked down upon what Allah (swt) has honored, and honored what Allah (swt) has abased.” (Wasa’il al-Shi’ah Vol. 6, Pg. 170, H. 3)