All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad, sallallaahu 'alayhi wa sallam, is His slave and Messenger.
You should remove whatever you find on the Mus-haf (the Quran hardcopy) even if you don't own it, on the condition that you don't cause any harm or damage to other people's property.
Shu’ab Al-Eemaan, authored by Al-Bayhaqi, stated when addressing the methods of glorifying the Quran:
“Among them is that one should not put another book on top of the Mus-haf, a garment, or anything else, unless one has two Mus-hafs, in which case, it is permissible to place one on top of the other.” [End of quote]
Allah knows best.