I think that particular criticism is vague and ill-formed, but oft-repeated because it sounds good. After all, having a brain hasn't stopped us from learning about the brain, even though one might think it would be hard to learn about brains with brains.
How do you know there isn't something special going on? Seems to me that believing there isn't anything special is just as much a guess as believing there is, given how little we understand consciousness.
So. Your title is misleading & not representative of the article, & that's annoying.
Although the hypothesis that the brain is a quantum computer is biologically and computationally implausible, there might be psychological phenomena not amenable to a neurocomputational explanation that are explicable by appeal to quantum theory. Penrose (1994, 598 A. Litt et al./Cognitive Science 30 (2006) 1997) and Hameroff (1998a, 1998b) argued that mathematical thinking and conscious experience are two such phenomena.
We argue that computation via quantum mechanical processes is irrelevant to explaining how brains produce thought, contrary to the ongoing speculations of many theorists. First, quantum effects do not have the temporal properties required for neural information processing. Second, there are substantial physical obstacles to any organic instantiation of quantum computation. Third, there is no psychological evidence that such mental phenomena as consciousness and mathematical thinking require explanation via quantum theory. We conclude that understanding brain function is unlikely to require quantum computation or similar mechanisms.
