All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
The Prophet said: “Do not greet the Jews and the Christians before they greet you; and when you meet any one of them on the road, force him to go to the narrowest part of it.” [Muslim]
According to the view of the majority of scholars of Islam, it is not permissible to take a lead in saluting Thimmis (free non-Muslim subjects living in a Muslim country). Such a ruling is with greater reason to be applied on disbelievers. But, if a disbeliever greets a Muslim, the latter has to respond with: "And on You", as reported in a sound Hadeeth.
The above ruling is related to taking a lead in saluting non-Muslims with “As-Salaamu 'Alaykum.”
As for greeting them with other forms except “As-Salaam”, the two schools of Ash-Shaafi'i and Ahmad also prohibit such a way. But Abu Haneefah and Malik and some scholars of the first two schools (Ash-Shaafi'i and Ahmad) are of the view that it is disliked to greet non-Muslims unless there is some genuine reason, such as a need for something from a non-Muslim, being a neighbor, or a relative.
As a final word, we state – by Allah's will – that the last point of view is the closest to what is right and correct.
For more benefit on greeting non-Muslims, please refer to Fataawa 88513 and 83602.
Allah knows best.