The duty of the Muslim towards a non-Muslim includes a number of things: 

Firstly: 

Da‘wah or calling him to Allah, may He be glorified and exalted. He should call him to Allah and explain to him the reality of Islam when possible, with regard to whatever issues he has knowledge about, because this is the greatest kindness that he can give to his fellow-citizens and to those whom he meets of Jews, Christians and others who may be mushrikeen (polytheists), because the Prophet (blessings and peace of Allah be upon him) said: “The one who guides others to goodness will have a reward like that of the one who does it.” And he (blessings and peace of Allah be upon him) said to ‘Ali (may Allah be pleased with him), when he sent him to Khaybar and instructed him to call the Jews to Islam: “By Allah, if Allah were to guide one man through you, that would be better for you than having red camels (the best kind).” And he (blessings and peace of Allah be upon him) said: “Whoever calls others to right guidance will have a reward like that of those who follow him, without that detracting from their reward in the slightest.” 

So calling him (the non-Muslim) to Islam, conveying Islam to him and being sincere towards him in that are among the best means of drawing close to Allah. 

Secondly:  

He should not wrong him, with regard to his physical wellbeing, his wealth or his honour. If he is a dhimmi (non-Muslim living under Muslim rule), musta’man (one who is granted security in a Muslim land) or mu‘aahid (one with whose country the Muslims have a peace deal), then he should give him his due rights, and not transgress against his wealth by stealing, betraying or deceiving, and he should not harm him physically by striking or killing him, because the fact that he is a mu‘aahid or dhimmi, or musta’man, means that he is protected by sharee‘ah. 

Thirdly: 

There is no reason why we should not interact with him, buying, selling, renting, hiring and so on. It is narrated in saheeh reports that the Messenger of Allah (blessings and peace of Allah be upon him) bought from kuffaar who were idol worshippers, and he bought from the Jews, and these are interactions. When he (blessings and peace of Allah be upon him) died, his shield was being held in pledge by a Jew for some food he had bought for his family (blessings and peace of Allah be upon him). 

Fourthly: 

With regard to greeting, the Muslim should not initiate the greeting, but he may return it, because the Prophet (blessings and peace of Allah be upon him) said: “Do not initiate the greeting of salaam with the Jews or Christians.” And he (blessings and peace of Allah be upon him) said: “If the people of the Book greet you with salaam (by saying al-salaamu ‘alaykum), say ‘Wa ‘alaykum.’” So the Muslim should not initiate the greeting to a kaafir, but if the kaafir initiates it, and the Jew or Christians etc. greets you with salaam, then you should say “wa ‘alaykum,” as the Prophet (blessings and peace of Allah be upon him) said. 

These are some of the rights between a Muslim and a kaafir. 

Another right is being a good neighbour. So if he is a neighbour, be kind to him and do not annoy him; give charity to him if he is poor, give him gifts, give him beneficial advice, because these are things that will attract him to Islam and to become Muslim; and because the neighbour has rights. The Messenger (blessings and peace of Allah be upon him) said: “Jibreel kept urging me to be kind to my neighbour until I thought that he would make him my heir.” Saheeh - agreed upon. If the neighbour is a kaafir, he still has the rights of a neighbour; if he is both a relative and a kaafir, then he has two rights: the rights of a neighbour and the rights of a relative. 

One of the rights of the neighbour is that you should give him charity, but not zakaah, if he is poor, because Allah says (interpretation of the meaning): “Allaah does not forbid you to deal justly and kindly with those who fought not against you on account of religion nor drove you out of your homes. Verily, Allaah loves those who deal with equity” [al-Mumtahanah 60:8]. According to the saheeh hadeeth narrated from Asma’ bint Abi Bakr (may Allah be pleased with her), her mother, who was a mushrik, entered upon her during the truce between the Prophet (blessings and peace of Allah be upon him) and the people of Makkah, seeking help. Asma’ asked the Prophet (blessings and peace of Allah be upon him) for permission - should she uphold ties of kinship with her? The Prophet (blessings and peace of Allah be upon him) said: “Uphold ties of kinship with her.” 

But with regard to celebrating their festivals, the Muslim should not take part in celebrating their festivals, but there is nothing wrong with offering them condolences if a loved one dies, such as saying “May Allah compensate you in your loss” and other kind words. But he should not say “May Allah forgive him” or “May Allah have mercy on him” if the deceased was a kaafir, and he should not pray for the deceased if he was a kaafir. But he may pray for the one who is alive to be guided and to be compensated and so on. End quote. 

Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allah have mercy on him). 

Fataawa Noor ‘ala al-Darb, 1/289-291.