The nuns, aged between 75 and 82, were discovered on Sunday after an armed robbery went wrong at a parish in the northern suburb of Bujumbura.
Police have detained two people for questioning.
The nuns were from the Italian Catholic diocese of Parma.
In a statement it said their deaths seemed to be "the tragic outcome of an armed robbery by a mentally unbalanced person".
Vatican Secretary of State Cardinal Pietro Parolin said: "Pope Francis has learned with great sadness of the murder of three nuns."