Does dark matter cause diffraction of light? If there is dark matter (Degrasse Tyson likes to call it "dark gravity"), would it diffract light?  If so, how could this be differentiated from gravitational lensing?  By spectroscopy? 
 A: No, dark matter is electrically neutral, so it doesn't interact with electromagnetic waves.
A: If you are asking if light is deflected by the gravity of dark matter, the answer is a definite "yes":  that is why gravitational lensing occurs.
Often we think of diffraction as  being an effect that requires periodic structures or abrupt changes in the phase and/or amplitide of a wavefront,  but the meaning of diffraction is more general than that.  Diffraction can be described as propagation of a wavefront,  in situations where Huygens' principle produces more useful results than ray optics.  Anything that modifies different portions of a wavefront differently will result in diffraction, and the same is true regardless of the mechanism of modification - so it is true even if dark matter's gravity is responsible.
You also asked how diffraction by (the gravitational effects of) dark matter could be differentiated from gravitational lensing.  The simple answer is that there isn't a distinction between the two cases.  
Perhaps more to the point is the fact that light following different paths around a clump of dark matter, even if the light arrives at the same place, won't form a static  interference pattern because the different paths are very unlikely to be the same length, and are sure to be changing continuously.  They can easily differ by light years when the dark matter clump is associated with a galaxy or cluster of galaxies.
