I personally find it hard to believe that the basic functions of neurons are directly dependent on quantum phenomena. Neurons and cells are HUGE compared to the single-atom spatial scales on which quantum mechanics plays a direct role in phenomena. So unless some crucial part of the neuron is very very small (on the order of a couple of atoms) it seems unlikely that quantum phenomena would have a direct function in the workings of the neuron. And if the brain had evolved functional information processing structures that small, how come we still have those awful energy intensify and bulky clumps of neurons sitting around? One would think that it'd be a relatively rapid evolutionary process to switch over to such small functional structures considering the advantages it would bring.
Secondly, it seems to me that any "harnessed" quantum effect would be very fragile. I don't know enough about this theory to say anything really informed about the feasibility of the specific models it's proposing, but it would seem that any phenomenon that microscopic would be extremely easily disrupted - wouldn't, say, a fluctuating external electromagnetic field be sufficient to collapse whatever fragile quantum state has been established in the brain? That'd make life hard on us cellphone users.
Of course, someone more knowledgeable may be able to confirm/debunk these criticisms. It would be interesting to see what other posters have to say.