This is a museum about the Whirling Dervishes. It's housed in a fine mansion, and shows displays.
It is part of a mosque's "külliye" (Islamic-Ottoman social complex centered around a mosque). It was built in the 17th century. The Mevlevi Lodge Monastery is entered via a courtyard which opens off the courtyard of the mosque.
Elsewhere I found it probably belonged to the Tekke Mosque, The mosque was built in 1638 by a Turkmen landowner, Mustafa Aga.