Here are some comments in the form of an answer, which can be viewed as the Koszul dual to Tyler's approach via the Eilenberg-Moore spectral sequence. 

Firstly, in the special case when $D$ is a point (or even contractible)  the map 
$$
A \to B \times C
$$ 
is a weak equivalence, so you can use the Künneth formula to compute the (co-)homology of $A$ interms of that of $B$ and $C$.

Secondly, if $D$ is path connected, we can choose a basepoint and pull everything back along the path fibration $\tilde D \to D$ to get a homotopy pullback
$$
\tilde A \to \tilde B
$$
$$
\downarrow \quad \quad \downarrow 
$$ 
$$
\tilde C \to \tilde D
$$
and because $\tilde D$ is contractible, the map $\tilde A \to \tilde B \times \tilde C$ is a weak equivalence.  To recover $A$ from this, note that $\Omega D$ acts diagonally on $\tilde B \times \tilde C$  (in some $A_\infty$-sense, but we can rigidify this if we want to using the Kan loop group). Taking  orbits we obtain
$$
A \to \tilde B \times_{\Omega D} \tilde C, 
$$
which is a weak equivalence (here the right side is really the Borel construction of $\Omega D$ acting on $\tilde B \times \tilde C$).

In particular, The above suggests that there's a quasi-isomorphism of DG coalgebras
$$
C_\ast(\tilde A) \simeq  C_\ast(\tilde B) \otimes_{C_\ast(\Omega D)} C_\ast(\tilde C) .
$$