I don't know how to make a follow up as opposed to a new answer, but this is a follow up to my preceding answer and the comments on it.
I believe that the commutativization functor C, defined by C(A) = A/[A,A], does NOT preserve weak equivalences, and I believe that Tyler is right and there are derived functors, but I don't know what they are.
Here is a simple example. Take the CDGA A=Q[x]/(x^2), where x is in degree 2. If I take a cofibrant replacement B for this in the category of DGAs, I will need a generator x in degree 2 and a generator y in degree 5 in order to kill x^2.
But then x^3 will be killed by both xy and yx, which are different elements in the tensor algebra on x and y that I have so far.
So I think I need a new tensor generator z in degree 6 to kill xy - yx.
When I apply commutativization C, xy-yx will go to 0, but z won't. So z will be a cycle that is not a boundary in C(B), so the map
from C(B) to A will not be a weak equivalence.
The answer to James's question is that C has to preserve weak equivalences between cofibrant objects, but not necessarily all weak equivalences, as apparently it does not.
So I have lots of questions myself about this now. Is it even true that the homotopy category of CDGAs is a full subcategory of the homotopy category of all DGAs? You think it must be, because if you start with a CDGA you can think of it as a DGA. But we have just determined that if you start with A, think of it as a DGA and take a cofibrant replacement B, then C(B) is not weakly equivalent to A. So you can't use this argument.