I think Keegan provides a great set of references, but I just wanted to expand on his answer in a little bit more detail. Penrose and Hameroff's ideas are mentioned a lot on the internet and although they are often debunked, you can never do it enough. I want to discuss (1) what microtubules are and (2) are there quantum effects in them? And, more importantly, (3) does that even matter? and (4) is any of this new?
What are microtubules?
Microtubules are a structure in the cytoskeleton that are found in all dividing eukaryotic cells and in most differentiated cell types (Desai & Mitchison, 1997). They are by no means exclusive to neurons, so if you think they bestow consciousness then you have to grant consciousness to all eukaryotes (something that most philosophers would find strange), or you are back at the same difficulty as before in trying to explain how networks of microtubules give rise to consciousness and thus postulating them as your basic units instead of neurons gives you no explanatory power.
Which organisms have the neuroanatomy Roger Penrose supposes play a role in consciousness?.
Are there quantum effects in microtubules?
In the video, Hameroff describes quantum and classic effects as a ying-and-yang. This is a very misleading picture, for a physicist the world is fundamentally quantum but at large sizes, and high temperatures (i.e. a lot of interaction with an external environment) is well described by our more intuitive laws of classical physics. Thus, the question isn't are there quantum effects, but are the quantum effects significant enough to cause non-classical consequences? Note, that most of chemistry can't be properly explained by classical physics, so every chemical reaction needs quantum effects to make sense, but we don't suggest that this makes every chemical reaction conscious.
The exciting part is that quantum effects do matter in (a non-trivial way) in some biological systems; most notably in photosynthesis (Engel et al, 2007). In particular, a certain energy perturbation after a photon is absorbed follows a quantum random walk, and one could speculate that this can be used for quantum computing, but there is no reason to expect it.
Microtubules are small enough to not completely rule out quantum effects. They are rope like polymers that grow to a length of about 25 micrometers (25000 nm), and have an outer-diameter of around 25 nm or about 200 atoms across. Researchers commonly use quantum dots to play with quantum effects, and these are typically spheres on the order of 10 to 50 atoms in diameter. Note, that we don't know how to couple 5000 quantum dots in one coherent chain (how many you would need to get the length of a microtubule).
However, the issue physicists usually raise, is not one of size but of the time that the microtubule would need to maintain coherence (i.e. a pure quantum state). The reason photosynthesis uses a quantum random walk, is because a classical one would not be fast enough to find a binding site. Thus, in that case the timescales involved are miniscule. The timescales involved in the function of microtubules are much longer, and physicists believe that their state is not coherence for that length of time (Tegmark, 2000).
Even if microtubules are quantum, so what?
This is the real clincher, suppose quantum effects are important to microtubules. Suppose that whole networks of microtubules are able to keep coherent entanglement between. Heck, suppose that the brain is a giant quantum computer. So what?
For a lot of people (like Hameroff), there is this misconception that classical physics does not allow for free well but quantum mechanics does because of wave function collapse. Thankfully, we have Conway & Kochen's (2006) free will theorem which says (more or less), any free will you give to an observer/experimenter in your philosophy of quantum mechanics, you will also have to give that some amount of agency to electrons and other subatomic particles. In other words, if your interpretation of quantum mechanics somehow gives you free-will then it is a trivial kind of free will that every particle in the universe has. Of course, this argument can be explored in much more depth, but I suggest reading someone that is familiar with quantum computing (Aaronson, 2013) instead of Hameroff.
For Penrose (see The Emperor's New Mind) consciousness is non-algorithmic and he suggests that a magical quantum computer could do these non-algorithmic tasks. The reason I use 'magical' is because a real quantum computer is Turing-complete, if a classical computer cannot solve a problem then neither can a quantum one (of course, if a classical computer can solve a problem, then quantum one can as well and might be able to do it qualitatively faster). For a nice computer science debunking of this part of Penrose's argument take a look a Scott Aaronson's lecture notes.
Is any of this even new?
For me, the most disappointing part of this pseudoscience is its lack of novelty. The basic philosophical urge underlying Hameroff's speculation is an over-application of reductionism: consciousness can't be something emergent, there has to be a basic essence to it. Since he can't take the dualist stance of Descartes, he instead postulates that it is"quantum magic". Even this use of quantum magic isn't original—it was an extremely popular type of hokum in the 70s that has been called quantum mysticism.
What makes people easily subscribe to pseudoscientific theories?
Aaronson, S. (2013). The Ghost in the Quantum Turing Machine. arXiv preprint arXiv:1306.0159.
Conway, J., & Kochen, S. (2006). The Free Will Theorem. Foundations of Physics, 36(10): 1441–1473.
Desai, A., & Mitchison, T. J. (1997). Microtubule polymerization dynamics. Annual Review of Cell and Developmental Biology, 13(1): 83–117.
Engel GS, Calhoun TR, Read EL, Ahn TK, Mancal T, Cheng YC et al. (2007). Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature, 446(7137): 782–6.
Tegmark, M. (2000). Importance of quantum decoherence in brain processes. Physical Review E, 61(4): 4194–4206.