In vitro reconstituted active systems, such as the adenosine triphosphate (ATP)-driven microtubule bundle suspension developed by the Dogic group [T. Sanchez, D. T. Chen, S. J. DeCamp, M. Heymann, and Z. Dogic, Nature (London) 491, 431 (2012)10.1038/nature11591], provide a fertile testing ground for elucidating the phenomenology of active liquid crystalline states. Controlling such novel phases of matter crucially depends on our knowledge of their material and physical properties. In this Rapid Communication, we show that the shear viscosity of an active nematic film can be probed by varying its hydrodynamic coupling to a bounding oil layer. Using the motion of disclinations as intrinsic tracers of the flow field and a hydrodynamic model, we obtain an estimate for the shear viscosity of the nematic film. Knowing this now provides us with an additional handle for robust and precision tunable control of the emergent dynamics of active fluids.