Let $A$ and $B$ be $A_\infty$-algebras. It's true, but it's a quite nontrivial fact, that the tensor product $A \otimes B$ can be given the structure of $A_\infty$-algebra, too. What is much easier to prove is that the tensor product of an $A_\infty$-algebra with a dg-algebra is again an $A_\infty$-algebra.

One can ask similarly whether the tensor product of a $C_\infty$-algebra and an $L_\infty$-algebra is an $L_\infty$-algebra, generalizing the simple fact that the tensor product of a commutative algebra and a Lie algebra is again a Lie algebra. But in fact I can't even prove what should be a much simpler statement: that the tensor product of a $C_\infty$-algebra with a Lie algebra is an $L_\infty$-algebra. Is this true?

Remark. What is quite clear is that the tensor product of a commutative algebra and an $L_\infty$-algebra is an $L_\infty$-algebra.