To kiss the Qur'an and then place it between one's eyes is an act of worship, through which one seeks closeness to Allah (swt). And acts of worship only become legitimate when irrefutable proofs establish their foundations. There is no such proof for the practice in question, but we do have a proof that we do not have doubts about, in which the Prophet (saws) said,
"Whoever introduces into this affair of ours that which is not from it, it is rejected"
The Companions performed only those acts of worship that they learned from the Messenger of Allah (saw). When 'Umar (r.a) saw the Black Stone he said "You neither cause harm nor benefit, and had not the Messenger of Allah kissed you, I would not have kissed you". When Mu'aawiyah kissed all corners of the Ka'bah, Ibn 'Abbaas (r.a) reproached him for adding to the practice of the Prophet (saaws). And when Al-Musayyib saw a man performing much Rukoo' (bowing) and prostration after Fajr Prayer, he censured him. The man said "O Aba Muhammad, will Allah punish me for praying?" Ibn Musayyib said, "No, but for going against the Sunnah (which is to pray directly after Fajr prayer until after the sun rises)."
The Permanent Council (of Scholars in Saudi Arabia) said: We know of no proof that establishes the legislation of kissing the Nobal Qur'an, which was revealed to be recited, to be contemplated, and to be acted upon.
[Fatwa # 8852 (4/122)]