We can see a number of ways in which the nature of German Christianity during the period both helped actively lead to and passively allow for the persecution, and eventually the extermination, of the Jews in Europe. In many cases, certain principles and perspectives derived from Christian tradition and thought actually made the Holocaust possible.
First, Christianity in Germany was, primarily, Lutheran in nature. Luther's influence on Germany is vast and this impacted the "Final Solution" in the following two ways: the Two Kingdoms doctrine & his Christian anti-semitism. Theologically, the German religious thinkers had created volkisch Christianity which viewed the German people as a/the source of authority. Finally, religious anti-semitism was melded to an economic-political conspiracy theory which focused on the Jews as the root of evil. These are just a few of the ways that German Christian culture effected the "Final Solution".
Luther's Two Kingdoms
The less discussed of these two is Luther's perspective on the relationship between the heavenly and the earthly, specifically in how Christians relate to government. In Temporal Authority: To What Extent it Should be Obeyed, Luther divided the world into the Kingdom of God and the Kingdom of the world. Not only this, but politically, there were two governments. He wrote:
One must carefully distinguish between these two governments. Both must be permitted to remain; the one to produce righteousness, the other to bring about external peace and prevent evil deeds.
Additionally, this second kingdom, meant to bring external peace, was to be followed. While he did allow for some resistance, as in the example of the Emperor's ban on Luther's books, such resistance was limited only to accepting punishment. Hence, only the theological was outside secular power. Additionally, he argued that only through ordained authority was one to protect the rights of their neighbors.
This doctrine had a significant impact on how Germans viewed their government. Most Lutheran Germans were taught, and accepted, that loyalty to the government was of paramount importance in the practice of their faith. Such loyalty is exemplified in the famous oath that the German army gave to Hitler. Whatever their own personal beliefs, they had a responsibility to the leader of the government (although giving an oath directly to the leader instead of to the government was-to my knowledge-abnormal.) Ultimately, however, while this loyalty to the secular state was used to support Hitler's rule, it was not the cause of the support for Hitler's government. Instead, the two kingdom doctrine was used by nationalist theologians to buttress support for Hitler. We can determine that they merely used Luther's doctrine because the same theologians had questioned and undermined the Weimar government using other writings of Luther (I did a paper on this topic and I find it fascinating).
Hence, it is probably best to say that many German's could alleviate themselves of their responsibility to the Jews by using Luther's Two Kingdoms to support their inaction.
Luther's anti-semitism
Luther's anti-semitism is a poorly kept secret. He originally believed that Jews had avoided conversion because of the failures of the Roman Catholic Church. Furthermore, he thought that when they were proselytized by his followers, they would see God's grace and convert. When this did not occur, he turned his well-known vitriol against them. He wrote:
They are real liars and bloodhounds who have not only continually perverted and falsified all of Scripture with their mendacious glosses from the beginning until the present day. Their heart's most ardent sighing and yearning and hoping is set on the day on which they can deal with us Gentiles as they did with the Gentiles in Persia at the time of Esther. Oh, how fond they are of the book of Esther, which is so beautifully attuned to their bloodthirsty, vengeful, murderous yearning and hope. The sun has never shone on a more bloodthirsty and vengeful people than they are who imagine that they are God's people who have been commissioned and commanded to murder and to slay the Gentiles. In fact, the most important thing that they expect of their Messiah is that he will murder and kill the entire world with their sword. They treated us Christians in this manner at the very beginning through out all the world. They would still like to do this if they had the power, and often enough have made the attempt, for which they have got their snouts boxed lustily.
Truly very little more needs to be said about this other than that Luther's anti-semitism was religious. Conversion was, one assumes, an acceptable outcome for Luther.
Volkisch Christianity
Some might disagree with the importance of this one, but with your question assuming some sort of common Christian morality, I think it is applicable. If such a morality exists, then one could assume it would be found in the Bible.
Some German theologians, most in fact, had rejected authority of scripture. In fact, many theologians had completely rejected scripture as God's form of revelation. Hence, rather than read and understand the world through the lens of scripture, they read scripture through their own political and economic lenses (some would say that this is the only way scripture is ever read.)
This, in turn, led to the German volk being portrayed as the revelation of God. In this way, the volk took on a Messiah quality and the Fuhrer of the volk even more so. Paul Althaus and others believed that a government that did not meat the needs of the volk should be opposed just as Luther had called for disobedience to a government that forced false religion on its citizens. This clearly demonstrates that the German volk was seen as the religious identifier for many theologians in Germany.
Such beliefs demonstrate that German theology had, at least in some cases, created a religion that held the German nation up as the revelation of God (in that they were the lens by which God demonstrated his will) and the object of salvation (in that only through a renewal of Germany could Christianity be saved).
Thus, while claiming Christianity, there were many aspects which many Christians would reject.
Economic-Political Conspiracy Theory
/u/commiespaceinvader has written a good post on this here; however, for the time being:
Luther's above text can show how German Christian culture 400 years later might still be able to see the Jews as conspirator's extraordinaire. To what degree German's believed Hitler's worldview is debatable, but there were assuredly some, notably the German Christian's, who bought into much of it.
Essentially, Jews were connected to both Bolshevism and Liberalism/Capitalism. Exactly why this occurred is for another day. It is enough to note that they were and discuss how this was seen by Christian Germans.
Liberalism was fairly early (as early as Freidrich Stahl in the early 1800s) on seen by Christians in Germany as promoting an individualism that would destroy a society's morality. Hence, the Lutheran church was supportive of a monarch who would have the role of maintaining a traditional morality in its people. This alliance of the Church and State in Germany continued to the fall of the Wilhelmine Empire. In the 1920s, this lack of morality in liberal states was on display, at least such was the view of many Christians, in the Weimar Republic. Many theologians of Weimar Germany, viewed the Republic with distrust or even hatred. Hence, Jews had brought the sinful individualism of liberalism to Germany and this was corrupting the German volk.
Bolshevism, and socialism in many cases, was seen as godless. The atheism of the bolsheviks ensured Christian opposition in Germany. Furthermore, Bolshevism was also seen as focused on material possessions, which would also have a deleterious effect on the German volk. Since Jews were seen as being directors and supporters of the Bolshevik Revolution, they were again seen as a threat to the morality of Germans.
For such reasons as this, many German Christians found themselves committing horrendous acts for the purpose of ridding Germany of a corrupting agent that threatened the German people's very souls. As I often tell my students: People don't think to themselves that they want to go do evil today. Instead, they tell themselves that the current act, although despicable, is ultimately good.
Conclusion
I hope that you can see how this all ties in to create a form of Christianity that was tethered to a traditional Christian morality in some ways, but in others was untethered completely. This is because although the form of Christianity was moral (at least in their eyes), it had as its source of morality the German people itself, not a traditional Christian source (such as the Universal Church or Scripture).
If I was unclear or missed anything, feel free to ask follow up questions.
Sources
Luther, Martin. “Temporal Authority: To What Extent it Should be Obeyed.” In Luther’s Work: The Christian in Society II, edited by Walther I. Brandt, 92. Philadelphia: Fortress Press, 1962.
---"On the Jews and Their Lies." Internet Archives.
Erickson, Robert P. “The Political Theology of Paul Althaus: Nazi Supporter.” German Studies Review 9, no. 3 (1986): 547-567.
Erickson, Robert P. Theologians Under Hitler: Gerhard Kittel, Paul Althaus and Emanuel Hirsch. New Haven: Yale University Press, 1985.
Edit: Link added