We investigate the effects of polymer molecular weight (MW) on the flow of flexible polymeric solutions in an extensional flow produced in a microchannel. The relatively small length scale and high deformation rates lead to significant stretching of the flexible polymer molecules near the hyperbolic point. For high molecular weights (MW~10^6 to 10^7), the stretching produces two flow instabilities, one in which the velocity field becomes strongly asymmetric, and a second in which it fluctuates in time. The frequency and amplitude of the velocity oscillations depends both on the Deborah number and on polymer molecular weight. These instabilities do not occur for semi-rigid polymer solutions or Newtonian fluids.