DUTIES OF A MUSLIM TOWARDS A NON-MUSLIM- AN OBLIGATION
Every title comes with a responsibility. If any person is an head of state or country then he/she has got some responsibility. Similarly the title Muslim too comes with the responsibility. The responsibility which was given to the Messengers and Prophets of God is now on Muslim ummah. Guiding society by spreading the call of righteousness and detestful things, encouraging virtue, enjoining all that is good, helping the people, and observing the Islamic principles in all aspects of public life: these are the duty of individual Muslims as well as the community working as a unit. Let us remember that Prophet Muhammad صلى الله عليه و سلم spent the first 40 years of his life speaking the truth and became famous as Al-Amin (the truthful) before he started preaching the Truth.
It is really important that Muslims present a model and an ideal way of life which may make them interested in studying Islam and eager to know the source of your guidance which enabled you to follow a particular way of life and worldview. This is the only way in which you can play an effective role in non-Islamic societies. There are many commands, and many prohibitions, and it is impossible to list them or explain them all in one place.
Allah gives command to Muslims in several places to convey the message of Islam.
Allah said: "Invite people to the way of your Lord, with wisdom and beautiful preaching; and argue with them in ways that are the best and most gracious: For your Lord knows best who have strayed from His Path, and who receive guidance." Qur'an.Surah Nahl 16:125
Except for those who have believed and done righteous deeds and advised each other to truth and advised each other to patience. Quran.Surah Asr 103:3
Say (O Muhammad): This is my way; I invite unto Allah (i.e. to the Oneness of Allah — Islamic Monotheism) with sure knowledge, I and whosoever follows me (also must invite others to Allah, i.e. to the Oneness of Allah — Islamic Monotheism with sure knowledge). And Glorified and Exalted be Allah (above all that they associate as partners with Him). And I am not of the Mushrikoon (polytheists, pagans, idolaters and disbelievers in the Oneness of Allah; those who worship others along with Allah or set up rivals or partners to Allah)” Qur’an.Yoosuf 12:108
You (believers) are the best of peoples evolved for mankind, enjoining what is right forbidding what is wrong and believing in Allah. Quran.Surah A’le Imran 3:110
We cannot get away with merely affirming that we are Muslims and that we have accepted Allah سبحانه و تعالى as our only God and Islam as our religion. Rather, as soon as we acknowledge Allah سبحانه و تعالى as our only Lord and His guidance as our way of life, we take upon ourselves certain obligations and duties towards ourselves and towards others.
Calling people to Allah سبحانه و تعالى is a virtuous deed which brings great reward. The Prophet Muhammad صلى الله عليه و سلم said: “Whoever calls others to guidance will have a reward like the rewards of those who follow him, without that detracting from their reward in any way. And whoever calls others to misguidance will have a burden of sin like the burden of those who follow him, without that detracting from their burden in any way.” Sahih Muslim, 2674
The first and foremost thing for any work is having knowledge about that subject. Similarly for conveying the message of Islam you should have knowledge about it. It is essential that the one who wants to call people to Islam should have some understanding of that to which he is calling them, but he does not have to know the entire religion, because The Prophet Muhammad صلى الله عليه و سلم said “Convey from me, even if it is one ayath.” Vol 4: 667
Da‘wah or calling Non-Muslims to belief and worship Allah سبحانه و تعالى is a duty assigned by our creator i.e, Allah سبحانه و تعالى. He should call him to Allah سبحانه و تعالى 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 Muhammad صلى الله عليه و سلم said: “The one who guides others to goodness will have a reward like that of the one who does it.” Abu dawud 5110
And he صلى الله عليه و سلم said to ‘Ali رضي الله عنه, 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 صلى الله عليه و سلم 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.” Sahih Muslim 2674
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.
There is no reason why we should not interact with him, buying, selling, renting, hiring and so on. It is narrated in sahih 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 ( صلى الله عليه و سلم) 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). 
These are some of the rights between a Muslim and a Non-Muslim.
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 صلى الله عليه و سلم said: “Jibreel kept urging me to be kind to my neighbour until I thought that he would make him my heir.” Sahih – agreed upon.
If the neighbour is a Non-Muslim, he still has the rights of a neighbour; if he is both a relative and a Non-Muslim, 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 سبحانه و تعالى says (interpretation of the meaning): “Allah 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, Allah loves those who deal with equity” Qur’an.surah al-Mumtahanah 60:8
According to the sahih 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 Muhammad صلى الله عليه و سلم and the people of Makkah, seeking help. Asma’ asked the Prophet Muhammad صلى الله عليه و سلم for permission – should she uphold ties of kinship with her? The Prophet Muhammad صلى الله عليه و سلم said: “Uphold ties of kinship with her.”
Abd al-‘Azeez ibn Baaz رحيمه الله said, with regards 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 Non-Muslim, and he should not pray for the deceased if he was a Non-Muslim. But he may pray for the one who is alive to be guided and to be compensated and so on. Shaykh ‘Abd al-‘Azeez ibn Baaz رحيمه الله. Fataawa Noor ‘ala al-Darb, 1/289-291.
Shaykh Ibn ‘Uthaymeen رحيمه الله said: If a person understands what he is calling people to, it makes no difference whether he is a great and prominent scholar or a seeker of knowledge who is serious in his pursuit thereof, or a regular person who has certain knowledge of the issue in question. The Messenger (May Allah honour Him and grantHim peace)said: “Convey from me, even if it is one ayath,” and he did not stipulate that the daa’iyah (caller) should have reached a high level of knowledge, but it is essential that he should have knowledge of that to which he is calling people. But calling out of ignorance or calling based on emotion is not permissible. Fataawa ‘Ulama’ al-Balad al-Haraam, p. 329