Is the carbanion formed in benzyne mechanism non-planar?
I have given one of these molecules a little spin on the computer. With whatever I tried, I was unable to find a non-planar solution. I really tried my best. Below is an animation (sorry for the poor quality) of an attempt which uses a chair like starting geometry. If there was any non-planar geometry lower in energy, this would have found it. 



The level of theory is DF-BP86/def2-TZVPP and the program used is Gaussian 09 rev. D.01.
There is really not much more I can add to the discussion. 


  I was told by my chemistry teacher that the carbanion formed in the picture above is non-planar. I don't understand why. It looks like benzene with an extra electron pair at one of the carbons so shouldn't it be planar?


To be absolutely, brutally honest here: I have no clue why your teacher had this idea. If this wasn't a "naked" carbon, we could talk about it, but it is essentially a deprotonated phenyl ring, and as such, you are absolutely right in assuming the lone pair being in phase with the ring. The &pi; system is hardly different from benzene. To fill this post a little bit more, have those fancy &pi;-orbitals.


(I could not find the completely anti-bonding pi-orbital.)

And for good measure, here is the HOMO, which is the lone pair at the carbon: