Does Hochschild (co)homology preserve quasi-isomorphisms? I.e. if we have an algebra in chain complexes $A$ and a chain complex $M$ that is a bimodule over $A$, we may form the cyclic simplicial object $\operatorname{cycB}(A,M)$, as we do for usual Hochschild (co)homology, and then realize it. Does this realization $|\operatorname{cycB}(A,M)|$ preserve quasi-isomorphisms of A and of M that respect the algebra and bimodule structures?
I am particularly interested in the case that $A$ and $M$ are formal, i.e. quasi-isomorphic through structure preserving maps to their homologies.
My thought is that it should because usually bar constructions come up as derived tensor products, however, I am in a situation where this would seem to lead to a contradiction, so I am perplexed.
I know if $A$ is commutative and $M$ is a commutative bimodule, we should be able to compare spectral sequences computing the Hochschild (co)homology (getting this from Weibel), but I'm not sure what happens in the non-commutative case.