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Yes, there is a nice way. The mapping cone $C_i^*$ is the homotopy pushout of $\ast \gets A_i^* \to B_i^*$, and $-\otimes X$ is a monoidal left Quillen functor (symmetric if the underlying ring is commutative), so $-\otimes^L -$ commutes with homotopy colimits. Hence, $C_1^* \otimes^L C_2^*$ is the mapping cone of $A_1^* \otimes^L A_2^* \to B_1^* \otimes^L B_2^*$. Derived functors and homotopy colimits can be computed via projective resolutions, so you have very classical inductive formulas to compute these quantities if you like.