Muslims should respect the Quran because according to Quran it is not a creature and or creator but Quran is word of Allah (کلام الله). Only respect it.
Also it is recommended to do not put Quran in the room you sleep or it is said do not put other book on Quran.
It is said by some scholars do not read Quran while sleeping. but none of them are Wajib (Obligatory) and are for Muslim's want to reach higher spiritual rank and become a close friend of God and meet God and even died before death (optional death).
Muslim can understand how to respect Quran on their own.
Let me say the memory of one scholar:
He said one day I went to a Great Scholar (Ayatollah Ghazi) and asked
him: I have heard some people when read Quran they can see the
paradise. why I can not? he replied: they do not read Quran while
sleeping and hold Quran with their both hands. I wondered how he knew I read Quran while sleeping and only
myself was aware of it.
When asked about the proper method to show respect for the Qur'an, Dr Liyakatali Takim made the following fatwa:
According to Ayatullah Seestani, we are not allowed to touch the
actual script of the Qur'an with any part of our body without
performing ablution. There is no harm in touching the translation of
the Qur'an in any language without wudhu.
You may read the Qur'an without wudhu if you do not touch the scripts.
So when a lady is in a state of haidh, she is allowed to recite the
Qur'an but not touch the words. It is makruh (but not forbidden) to
recite more than 7 verses when in haidh or Janabat.
Ayatullah Seestani also adds that based on obligatory precaution, it
is not allowed to touch the name of Allah without wudhu, regardless of
the language it may be written in. It is also better not to touch the
names of the Prophet, the Imams and Bibi Fatima without wudhu also.
See Islamic laws explained p. 56
Ayatullah al-Khu'i further adds that even the attributes (sifat) of
Allah should not be touched without wudhu. See MInhaj vol. 1, pg 50
Also the Qur'an should be treated with extreme respect, no act should
be done which the urf (masses) may consider violating the sanctity of