Can Muslims pray for the forgiveness of non-Muslims? The following verse apparently leaves no room for this:
It is not (proper) for the Prophet and those who believe to ask Allah’s Forgiveness for the Mushrikūn even though they be of kin, after it has become clear to them that they are the dwellers of the Fire.(9:113)
In principle, we can pray for any person in a general way by asking God to grant him excuse for his sins if he deserves this rebate and to grant him a place in Paradise if he deserves this reward.
As far as the Qur’ānic verse that you have cited is concerned, it specifically pertains to the Idolaters of Arabia. I have explained several times before in these columns that there are certain directives of Islam which are specific to the age of the Prophet (sws) and his companions. In the age of Messengers of Allah, the Almighty punishes people who persist to deny the basic truths which He unveils through His messengers so that this exercise can become a visual evidence to the reward and punishment that is going to take place in the Hereafter. In the age of the Messengers, people who deliberately deny the basic truths can be pinpointed because it is the Almighty who informs the Messengers about this. The punishment for denying the truth in spite of being convinced about it begins in this very world. Now asking for forgiveness for people about whom it is known that they have been condemned to Hell, is obviously not required. The last part of the verse clearly says that the real reason for abstention from such a prayer is the fact that their fate has become evident because of their obdurate and deliberate denial.
Today also people may deny a truth in spite of being convinced about it but since we can have no knowledge of this (because this can only be obtained through divine revelation which, of course, no longer comes), so, in my opinion, we can pray for any Muslim or non-Muslim in the above mentioned general way.