At first there was some dialogue as Protestants 'discovered' Eastern Orthodoxy, made contact, and both sides tried to work out whether they could be friends, and whether they might integrate. It became clearer that the Orthodox had little interest in 'reforming', and the Reformers were not going to simply assimilate into Orthodoxy.
This is a copy-paste from my own comment to a similar question.
During the reformation there was correspondence between Protestant theologians and the Orthodox churches. For example, in 1559 Philip Melancthon wrote a circular letter indicating that he had sent a Greek translation of the Confession of Augsburg to the Patriarch of the Byzantine church. This letter is carried by Demetrios who has spent time in Wittenberg having been dispatched by the Patriarch in order to get information on the Reformation. Melanchthon received no reply, but the Tübingen theologians made contact with the patriarch Jeremias II. Again the Patriarch is sent the Greek translation of the Augsburg confession. This is 1574, and he responded with a committee of Orthodox theologians to look at it and write a response. It is not highly critical of Lutheran doctrine, but it maintains an Orthodox line. In 1577 the Lutherans send a reply, highlighting 3 particular points of difference, particularly regarding Free Will, the place of Works, and the relationship between Scripture and Tradition. A second reply from the Patriarch comes in 1579 upholding the Orthodox position especially regarding church tradition and against the Reformation reliance on Scripture alone.
In the early 17th century a certain Cyril Lucaris became Patriarch of Alexandria. He had been involved with Jeremias II and so had some knowledge already of the interactions with the Lutherans. Then from 1620 he was elected as Ecumenical Patriarch, a position he held five times (being also deposed five times). Cyril Lucaris appeared sympathetic to Protestant doctrines, especially when Protestants published a "Lucarian Confession" in 1629 in Latin, in which the Patriarch appeared to endorse Calvinism, and from 1629 to 1633 an "Eastern Confession of Christian Faith" was also published under his name in Latin, Greek, French, German, and English. Again, it is unclear whether Lucaris actually held these views, but he did not disavow them in the written evidence. Cyril also maintained correspondence with the Anglican church, and sent students to study in England, Switzerland and the Netherlands. However these initial contacts didn't really amount to anything. It became clear to both sides that there were fundamental divisions not just in theology, but theological presuppositions, and politically and culturally the Orthodox church was quite distant from Protestantism, even though in some particulars it was closer to Protestantism than Roman Catholicism. For these reasons the Reformation did not really impact the Eastern Church that much.
I will have to come back later to edit in a source, but I did a research paper in uni over the opening of Anglo-Russian relations, which occurred under the rule of Elizabeth. Part of my research was a brief overview of religious differences. Essentially, Tsar Ivan IV saw Protestant England as religiously backwards, while feeling that Eastern Orthodoxy was the superior religion. From my research, they accepted Protestantism because they found a mutual enemy in Roman Catholicism. Note that at the time, the Catholics in France and Spain (Portugal as well?) nearly waged war against England because of its departure from the Papacy. Similarly, the Eastern Orthodox Church had dealt with Catholic Crusades in the past, leaving tension between the Byzantines and the Catholics.
I know I'm only focusing on the Russian views; however, it seemed that the ideology was essentially a case of "the enemy of my enemy is my friend," and since the Eastern Orthodox and Protestant Churches had Rome as a common enemy, they were able to tolerate each other.
As I said, I'll edit in my sources in a little bit when I find my paper, but that's just a brief overview of what I know.