Leaving the Schengen area with a Refugee Travel Document issued by the USA

We've had a similar question with a relevant paragraph about your situation:

According to this official document (in French), among Schengen countries, US refugee travel documents are accepted for visa-free short term travel to Belgium, Croatia, Germany, Hungary, the Netherlands, and Slovakia.

However, if you enter Schengen via Germany and happen to take a short side trip to Italy this might not be a practical problem, provided you don't cross a passport control. While there are mostly no border controls within the Schengen area, there may be occasional random checks especially on public transport, and border guards in countries other than Germany may be unfamiliar with the situation. This case is so unusual that it's not really clear what would happen, though I would guess they would see that Germany admitted you and send you on your way. However, it is possible that you could be detained and removed from such a country as Italy.

If you simply visit a country like Denmark during your trip, it's possible that no one will check you at the border, thanks to the Schengen area. However when you try to leave at the airport, Danish immigration personnel will certainly know you've entered their country illegally and proceed to apply any number of penalties: from a mere warning to official deportation, including a temporary ban from the Schengen area.

In short, don't risk it. Applying for a Schengen visa should be relatively straightforward and it will save you the headache of worrying about violating potential laws. Having a used Schengen visa will also increase your trustworthiness when it comes to applying for other visas, such as UK or Irish ones. And when you finally receive your US passport you won't have any immigration violations in your history, so you can visit all of the mentioned countries without worries.