In Germany, are there any cities or special regions in cities which are dangerous for non-white people to go to?

No, anti-nationalist propaganda has been very strong in Germany ever since the end of WWII, so if anything, visibly nationalist people will have a hard time in many places.

Although places such as Marzahn in Berlin are notorious for the skinheads that used to operate there, these mostly did their thing in the background and the gangs have since largely disbanded.

As such, you'll be just fine across the country

Germany has very few areas where no non-white people live. Which probably implies that there are no areas that are really dangerous. That said, there are certainly areas that can be more unpleasant than others. Very generally spoken, the East is more racist than the West, small towns and villages are worse than bigger cities, socialist concrete appartment quarters are worse than more affluent neighbourhoods.

There are also some individual factors that contribute to how dangerous or unpleasant a place is. E.g. East Asians probably have less problems than Black people or people who look like refugees (given that every East German small town has some Vietnamese inhabitants and businesses). A 1.95m athlete is going to have less problems than a 1.70m shy guy. You are more likely to run into trouble when using public transport at night than when always travelling in your own car.

But as pointed out by the other answers, the actual risk of getting hurt is really low. The risk of someone shouting abuse is (in some areas) considerably higher, but still not so high that it would spoil your trip. Except if you plan on spending your whole trip in especially troublesome and non-touristy places.

Contrary to what Tor-Einar said, my impression is that the situation has actually steadily improved in the last few decades. The situation is certainly much better than in the 1990s.

Source: Having travelled a lot through East Germany (esp. Saxony) with people that look East Asian. And havng travelled a bit through Saxony with people that are (and look like) Middle-Eastern refugees.

For some concrete examples with touristy places that have a rather bad reputation: Dresden is fine to unless you want to be really cautious. I have heard from a refugee who lived in Bautzen for a few months that it was quite unpleasant. Though nearby places like Görlitz supposedly were OK. If you do not mind a certain probability that some idiot might be shouting nonsense at you, even visiting Bautzen should not be a problem. But you might think twice if you are travelling with people that take things like that to heart.