It is He Who has created you; and of you are some that are unbelievers, and some that are believers; and Allah sees well all that you do. (64:22)
A characteristic of this universe is that nothing has been created without its pair, i.e. everything has an opposite which sets up resistance to it. Sustenance of a system which consists of such opposing forces with all its complexities is the work of none other than the ultimate Supreme Being Who has the power to maintain this equilibrium. No doubt, each and every creation, is appreciable in its own right, but the real beauty lies in the balance of the grand design of this universe as whole.
Applying this fact to a human society, we find that the non believers set up resistance against believers. The nonbelievers become foil to the latter and make their presence significant—the benefit of light cannot be visualized without experiencing the pitch of darkness. The hardships endured by a Muslim resisting the unbelievers reflect his extreme love for God. He bears all severities in his divine love with the assurance that Allah is All-Seeing. The existence of night is important to highlight the benefits of day but this doesn't mean that night is of no value. As a matter fact, night has its own system (of course, highly important for all creatures on the earth). Similarly, in a human society the presence of unbelievers side by side with the momineen (believers) is inevitable. But (here) we are not required to be concerned for the logic behind the very existence of unbelievers.
For example, while walking, if we stumble over some stone, we try to find a better passage to reach our destination. A single hit of a stone makes us realize the need for a safer path. Whether this was the only good/utility of the stone? It is not for the traveler to stop and ponder over such questions; rather, he is required to (quickly) resume his journey towards the destination. The objective of the annunciation of the divine Prophets (May Peace and Blessings of Allah be upon all of them) is perfecting mankind in basic human values. Discussions on things other than humanity are out of the purview of their teachings. But this should not construe the misunderstanding that God has nothing to do with them. Certainly, He has created them all with His Own Logic behind each and every one of His creations. But, we do not dilate upon them, as all these are not directly involved in matters of human character, actions and interactions.
One relation of a born unbeliever with a Muslim is that of a human being. His very existence is an obstacle in a momin's way. A momin is supposed to remove it through his wisdom and determination. This shows his strength of faith. This much is directly related to divine laws. Secondly, the religious norms also envisage making efforts by a true Muslim with perfect faith to convince an unbeliever to embrace the true faith of Islam as he is a part of human fraternity and a momin feels it his duty, by virtue of his faith, to invite fellow human being(s) to the Path of Almighty Allah. A true momin acts from two aspects (and the Islamic laws dilate on both), i.e.
1). A non believer poses an obstruction in a momin's way who struggles to remove him to clear his way;
2). Humanity of an unbeliever is dull and the momin tries to brighten them.
Any other logic, whatsoever, behind the very existence of a non-Muslim is not the focus of discussion among experts of divine studies.