Luther spoke on this topic, developing the idea of a Kingdom of the Left (somewhat akin to secular authority) and a Kingdom of the Right (the Church). This was not, as some people conclude, an early version of separation of church and state. Rather, it was based on the idea of "vocation", that God calls each of us to a task and we should do our task, not meddle in the tasks of others.
More strictly speaking, the two kingdom theology posits that the left hand kingdom rules by Law, and this is the vocation and duty of the state to curb lawlessness; the Church on the other hand, the right hand kingdom, operates solely by the Gospel.
While not strictly speaking a separation of Church and State in the modern sense, it was one of the inspirations for the principle of separation, James Madison (the father of the US Constitution) credits the Lutheran doctrine of the two kingdoms as one of the influences on his thinking in this area. How influential I don't know, it's a subject I'd need to look into more.
I recently read Pastor Dietrich Bonhoeffer's essay The Church and the Jewish Question, in which he seeks to address the political and ecclesiastical issues of the 1930's and what the Church's response must be. It was written specifically following "the Aryan paragraph" which represented the kinds of inroads the Nazis had made into co-opting the Protestant Church of Germany (through the Deutschechristen movement), in which Christians of Jewish background were barred from the Church in spite of the fact that they were baptized Christians. Bonhoeffer begins by making clear the distinction between the state's duty of Law and the Church's duty of Gospel, and that the Church cannot impose itself upon the state moralistically, so that even if the state is doing law and order wrongly, the Church cannot coopt the vocation of the state; though individual Christians as citizens through their vocation as citizens most certainly can voice themselves either as individuals or collectively through various humanitarian organizations--but the Church as Church cannot. However, Bonhoeffer continues that the Church might find itself in a position where it must act, through criticism and condemnation of the state, by consoling and tending to the wounds of the victims of the state, and if push comes to shove, to throw itself headlong and say enough is enough.
For Bonhoeffer enough was enough, by the imposition of the state upon the Church in telling the Church how to govern herself, by telling her that baptized Christians had no place in the Church because of their Jewish background the state has crossed a line that the Church cannot abide by. It is here that we find the Kirchenkampf, the Church-Struggle, between the Deutschechristen and the emerging Reichskirche, and its opposition in the underground Confessing Church which would come to produce the Barmen Declaration which asserts that the Church confesses the sole Lordship of Jesus Christ, and that the Church's authority is through the Gospel and from Christ through her ministers of the Word, and no other leaders (
fuhrers in the German) can be accepted. The imposition of the state upon the Church is wicked and wrong. And the Church must make its stand against the state for the state has not only deprived people of their due rights by its unjust laws against the Jews (and the Holocaust that followed obviously), but was forcing its hand against the Church; and thus the Church became morally obligated to speak and act, to not just bandage the wounded crushed beneath the wheel of injustice, but to take hold of the wheel itself.
It probably seems strange to some to see Bonhoeffer move carefully in discussing how the Church cannot impose itself moralistically upon the state, in order to eventually wrestle with these things theologically before arriving to a basis of action; but Bonhoeffer was a Lutheran and theology matters even in crisis. The Church must still be the Church no matter the situation, for the Church to lose her identity as the Gospel people, and instead think herself something else, is to cease to be the Gospel people and becomes either subservient to, or an arm of, the state.
Bonhoeffer's influence (at least it seems here) on MLK can probably also be seen here, as King would later say,
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And those who have gone to the church to seek the bread of economic justice have been left in the frustrating midnight of economic privation. In many instances the church has so aligned itself with the privileged classes and so defended the status quo that it has been unwilling to answer the knock at midnight. The Greek Church in Russia allied itself with the status quo and became so inextricably bound to the despotic czarist regime that it became impossible to be rid of the corrupt political and social system without being rid of the church. Such is the fate of every ecclesiastical organization that allies itself with things-as-they-are.
The church must be reminded that it is not the master or the servant of the state, but rather the conscience of the state. It must be the guide and the critic of the state, and never its tool. If the church does not recapture its prophetic zeal, it will become an irrelevant social club without moral or spiritual authority. If the church does not participate actively in the struggle for peace and for economic and racial justice, it will forfeit the loyalty of millions and cause men everywhere to say that it has atrophied its will. But if the church will free itself from the shackles of a deadening status quo, and, recovering its great historic mission, will speak and act fearlessly and insistently in terms of justice and peace, it will enkindle the imagination of mankind and fire the souls of men, imbuing them with a glowing and ardent love for truth, justice, and peace. Men far and near will know the church as a great fellowship of love that provides light and bread for lonely travelers at midnight." - Dr. Martin Luther King, Jr, A Knock at Midnight
-CryptoLutheran