If A is a commutative algebra and B is an X- algebra, then the tesnor product $A \otimes B$ is an X-algebra (so for example, $Com \otimes Lie$ is a Lie algebra). This is seen using the language of operads. Let Com be the commutative operad. Since Com(n) is a one dimensional vector space for every n, tensoring Com with an operad O doesn't change the operad O.

Does a similar thing hold true for a C-infinity algebra? That is, if A is a $C_\infty$ algebra, is $A \otimes B$ an $X_\infty$ algebra?

I'm still trying to familiarize myself with the language of operads, and perhaps the question can be made more precise in that language, where the infinity version of an operad is cofibrant resolution of the operad.