According to Young Germany:
Do I need a visa if I am not from Germany?
If you are an EU citizen, you don't need a work permit. If you are a
non-EU citizen, you do. If you are studying and living outside Germany
and applying for an internship in Germany, you will need to organize a
work permit before you are allowed to undertake an internship. You
will also need health insurance, which is compulsory for anyone
working in Germany, and necessary for getting a work permit.
Therefore the answer is yes, you will need a work permit before coming to Germany for work. However if your employer doesn't ask you to apply for one, you might as well get away with going to Germany on a regular visa, especially since the internship is unpaid.