Can I drink tap water in Germany?

Yes, tap water is the safest and most controlled beverage/food product in Germany. Many German cities including Berlin and Munich brag about the quality of their tap water which often comes from the same source as mineral water. 

In multiple blind-tests, various mineral water (still and sparkling) were tested against normal and carbonated tap water. Tap water came out at the top or equal to the mineral waters.

Like everywhere else the taste and quality of the tap water differ across the country. Usually, taste is related to the hardness of the water. While softer water tastes good almost everywhere, harder water generally tastes better cold. 

For those that simply don’t like the taste of the water, a faucet water filter such as Brita Filter which removes chlorine and odor is very efficient. In addition to improving taste, it also removes a further 80+ potential tap water contaminants, including byproducts of Chlorine, identified as cancerogenous. But that investment would only make sense if you're in Germany long-term.

Another tip is to simply pour the water into a carafe and leave in the fridge overnight whereasmost of the chlorine evaporates.