The Computerspielemuseum in East Berlin has already been mentioned, but there are also some old computers or reproductions (including a working reproduction of one of Konrad Zuse's first computers) in the Technikmuseum in Berlin. The Technikmuseum is in the Western part of Berlin, but if you happen to be in town anyway, you should not miss it.
Dresden was the center of East German computer production, and they seem to have some examples at the Technische Sammlungen, at the Robotron museum (HT to @scai) and at the Technical University. The latter two can only be visited after making an appointment.