بسم الله الرحمن الرحيم
(Fatwa: 259/229/SN=03/1437)
It is not lawful to touch the Holy Quran without wudhu. It is sin. You should do true taubah from this and should not repeat this again. The taubah will purge the sin away, there is no special kaffarah for this in the Shariah except taubah.
Allah knows Best!
Darul Ifta,
Darul Uloom Deoband
