Model structure of commutative dg-algebras inside all dg-algebras Most of the literature considers the standard model category structure on (graded) commutative differential algebras. But this generalizes to all (not-necessarily commutative) dg-algebras.
Details and references are listed here: nLab: model structure on dg-algebras.
What is known about how commutative dg-algebras sit inside all dg-algebras intrinsically ?
Do we have, for instance, a Quillen inclusion of the CommutativeDGAlgs into DGAlgs, such that its adjoint is a localization?
 A: OK, first of all we had better be working over a field of characteristic 0 if we expect commutative DGAs to be a model category where the weak equivalences are homology isomorphisms and everything is fibrant.  Otherwise it is impossible.  
In this case, it is true that the forgetful functor from commutative DGAs to DGAs preserves fibrations and weak equivalences, so is a right Quillen functor.  Its left adjoint of abelianization is therefore a left Quillen functor.  
However, it is definitely not true that commutative DGAs are a localization of DGAs by this adjoint pair; the map from a DGA to its abelianization is not going to be a weak equivalence, as it won't be a homology isomorphism.  
As I recall, an abelian object in the category of DGAs is not a commutative DGA; it is instead a square 0 extension.  So if you do Quillen homology (derived functors of abelianization) in DGAs, you are supposed to recover Hochschild cohomology of DGAs, I believe. 
On the other hand, it does seem to me that you could try to declare the map from a DGA to its abelianization to be a weak equivalence.  I don't know what would happen then.
                                    Mark Hovey
A: 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.  
A: By the adjoint functor theorem the inclusion has a left adjoint iff it preserves limits, which it does, if I am not completely wrong.
Weak equivalences and fibrations are exactly the same (defined via the underlying complexes) in both categories, so the inclusion functor does preserve fibrations and trivial fibrations, so you have a Quillen inclusion. 
The adjoint is a localization in the sense that you could equivalently localize the category of dg-algs, by declaring the maps from a dg-algebra to its abelianization to be an equivalence and inverting those. By the universal property of the localised category you have a functor to the category of commutative dg-algebras and this functor should be an equivalence. But here you have an actual localization not a Bousfield localization.
My impression is that you can take the model structure on dg-Alg and Bousfield localize it with the above additional weak equivalences. Conveniently everything is fibrant, so the local objects would be exactly the commuative dg-algebras. Probably this was your actual question, but I am not sure about this - better wait for a hotshot to confirm my guess or tear it into pieces...
A: I can second Mark's example, however I would point out that the cocycle z lies in degree 8 and not 6.  There's another one, w in degree 11 whose purpose is to kill y^2, which is 0 in the abelianisation.  Not to mention xz in degree 10.  The resultant commutative algebra seems to be of the form k[x]⊗k[V], where V has some kind of algebraic structure, I'm guessing a (co)lie-module after a suspension, but I can't work out what it should be.
I also have an interpretation of all of this.  In working out an associative quasi-free presentation (TW, δ) we're actually working out the bar homology W of the algebra (actually including the A-infinity coalgebra structure as well).  But there's a decomposition of the bar homology of a commutative algebra known as the λ-decomposition (see for instance Loday's Cyclic Homology book).  If we were to take the "middle" piece W' of this decomposition we would get the commutative bar homology (and it's associated L-infinity coalgebra structure).  And this is just what we need to get the resolution of the original algebra as a quasi-free commutative algebra (SW', δ').  All the extra bits (the V of the example) come from other pieces of the λ-decomposition.  I think that this should impose strong structural conditions on (SW, δ'').
Going back the example we can work out that z is in the second part of the decomposition, although I'm not sure about w.
So what this means for the various homotopy functors between the homotopy categories in question I'm not sure, the following is speculative:  It seems to me that it indicates that they're not full.  There are homotopy morphisms from k[x]/x^2 to another commutative algebra that are not homotopy commutative algebra morphisms.  They should fit into the λ-decomposition and their position there should indicate just how "not commutative" they really are.
Final note on "not commutativity": this notion can be made more rigorous using operads, but in a hand-wavey way it just means to what level the higher homotopies of the commutativity condition hold.
A: in the rational case, the forgetful functor U: dgCAlg-------> dgAlg induces a faithful functor in homotopy category under the following conditions.
R is connected (connective) differential commutative graded Q-algebra and S= C(X,Q) (the cochaine complex of a space with coefficients in Q, then
Ho(dgCAlg) (R,S)-----> Ho(dgAlg)(UR,US) is injective.   
