A friend of mine doesn't want to say 'Sallallahu Alayhi wa Sallam' when the name of the Prophet Muhammad (ﷺ) is mentioned. What's the ahkam in the Qur'an and Sunnah about this?

It is not waajib (obligatory) to make the du'a for the Prophet (ﷺ), he just misses out on the blessings from doing so. That is all.

However, it is good to find out from him, why he chooses not to do it, as he may be under a misunderstanding which you can clarify.