Mind Hacks has a post on a critique of mirror neuron research here.

While this new paper may contradict some of the more rigorous parts of the definition of mirror neurons, I would still consider it significant that neurons in motor areas are activated when viewing others' movements at all, especially if the neurons activated during observation and execution of the same action are in close proximity (even if not identical).

Also, it seems that any correlation of neural response to observed actions with measures of empathy, linguistic processing, etc. still stands. By that I mean, if activity in a certain brain region when viewing simple movements is correlated with any higher-level cognitive function, this points to a similar mechanism for the two regardless of whether true mirror neurons exist.

So essentially, it seems this is a debate over the specific neurophysiology of the response to observed movements, not whether motor imitation has given rise to anything else.