When is bar-cobar duality an equivalence?

What the references are saying is correct, and you are right. Yes, $\Omega BA \to A$ is always a quasi-isomorphism. No, $\Omega$ does not in general take quasi-isomorphisms to quasi-isomorphisms.

A sufficient condition for $\Omega$ transforming a DG-coalgebra morphism to a quasi-isomorphism of DG-algebras is a filtered quasi-isomorphism of conilpotent DG-coalgebras. This also generalizes to CDG-coalgebras (which correspond to nonaugmented DG-algebras, and for which the conventional notion of quasi-isomorphism does not even exist, but filtered quasi-isomorphisms make perfect sense).

For a reference, see my 2011 AMS Memoir "Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence", http://arxiv.org/abs/0905.2621 , Section 6.10.

See Hasegawa's thesis, available here. Lemma 1.3.2.3 is what you want, on page 35 of the pdf. To see that $\Omega B A \to A$ is a quasi-isomorphism, one filters $BA$ by its primitive elements (i.e. by the length of bar elements) which then induces on $\Omega BA$ a filtration. One gives $A$ the trivial filtration (i.e. $F^0A=0$ and all higher $F^i$ are $A$) and sees the counit is a filtered morphism. The associated graded map is the identity of $A$ on degree one, and then one shows that $\text{Gr}_i(\Omega BA)$ is contractible for $i>1$.

Another way to remember why the counit is a quasi-isomorphism is that it corresponds to the trivial twisting cochain $\beta:BA\longrightarrow A$ whose total space $BA\otimes_\beta A$ is acyclic. Then one fits the counit into the following sequence

$$1\otimes \varepsilon_A : BA\otimes_\gamma \Omega B A\to BA\otimes_\beta A$$

Both twisted complexes above are acyclic so $1\otimes \varepsilon_A$ is a quasi-isomorphism. Now $BA$ is simply connected so a spectral sequence argument works to show that, because $1$ is of course a quasi-isomorphism, so is $\varepsilon_A$.

This argument fails for $\Omega$ because it decreases the connectivity of your space. Lemma 1.3.2.2 in the thesis above gives a sufficient criterion for $\Omega$ to preserve quasi-isomorphisms. The bar construction, on the other hand, preserves quasi-isomorphisms by a direct argument using the filtration by length and the Kunneth formulas. Again, no connectivity issues arise here, which renders the desired spectral sequence convergent.