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 $PB \to B$ to get a homotopy pullback $$ \tilde A \to \tilde B $$ $$ \downarrow \quad \quad \downarrow $$ $$ \tilde C \to PD $$ and because $PB$ 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). We can then the evident map $$ \tilde A \to \tilde B \times_{\Omega D} \tilde C $$ 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 $$ C_\ast(\tilde A) \simeq C_\ast(\tilde B) \otimes_{C_\ast(\Omega D)} C_\ast(\tilde C) . $$

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 $$ C_\ast(\tilde A) \simeq C_\ast(\tilde B) \otimes_{C_\ast(\Omega D)} C_\ast(\tilde C) . $$