All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
It is more appropriate to hold the Qur'an with the right hand as a sign of honour to it. The Sunnah of the Prophet illustrated that it is desirable to use the right hand in everything that we wish to honour.
'Aa'ishah said: "That the Prophet used to prefer to start with the right hand or right side (over the left) when putting on his shoes, combing his hair, putting on perfume, and in all his matters." [Al-Bukhari and Muslim]
'Aa'ishah also said: "The Prophet used his right hand in applying perfume, in eating, and used his left hand in the toilet or when removing any harm." [Abu Daawood]
So the above narrations illustrate that it is preferable to use the right hand when honouring anything, and the left hand is used when there is any harm or dirt, like using the toilet.
Imaam An-Nawawi said: 'It is preferable to use the right hand (rather than the left hand) in anything that requires honour, like Wudu (ablution) Ghusl (having a bath to remove major impurity), putting on clothes, shoes, socks, trousers, entering the mosque, using Siwak, eyeliner (Kohl), etc.'
There is no doubt that using the right hand to hold or carry the Qur'an, has more importance than using it to put on socks.
Al-'Izz Ibn 'Abdus-Salaam, spoke eloquently about the desirability of using the right hand saying: 'There is no doubt that encountering what is honourable with what is honourable is something good (reasonable) in mind.'
Finally, we have to say, however, that we do not know of any person of knowledge who gave a Fatwa that it is impermissible or dislikeable to hold or carry the Qur'an with the left hand.
Allaah knows best.