I have heard from someone that on the day of judgement Quran will tell which person fulfilled its rights. So, I want to ask What are the rights of Quran on us ?
The only thing that comes to mind is Quran 25:30
"And said the Messenger, "O my Lord! Indeed, my people took this the Quran (as) a forsaken thing."
The word used is MAHJUR, which means to immobilize something. The villagers who bind a cow by tying her front foot to her horn they effectvely MAHJUR it.
For a Coranist interpretation of the question, we read in 2:256 (Pickthal) that
There is no compulsion in religion. The right direction is henceforth distinct from error. And he who rejecteth false deities and believeth in Allah hath grasped a firm handhold which will never break. Allah is Hearer, Knower.
Since this Coran is a (not necessarily strict) subset of this religion, it follows that there can be no compulsion with respect to the Coran, and hence it has no a priori rights on us. Furthermore, we read in 88:21-22 (Pickthal) that
Remind them, for thou art but a remembrancer,
Thou art not at all a warder over them.
We note that the entire chapter does not specify who is being referred to in the second person. The standard interpretations, assuming a difference between the Messenger and the Message being sent, conclude that it is the Messenger that is referred to in the second person. However, by relaxing the above assumption, we find that by regarding the Coran itself as a rasool, the verse has as a direct corollary that it has no rights over them ("thou art not at all a warder over them").