Let $X$ a smooth manifold. Is the pullback morphism $\Omega^\bullet(X)\to\Omega^\bullet(X\times \mathbb{R}^n)$ an acyclic cofibration of differential graded commutative algebras? I guess so, and even that this should be the basic example to have in mind, but being no expert, I don't trust myself too much.

It depends completely on what you mean by cofibrations. The choice is not quite simple to make as the homotopy category of real commutative dga's is antiequivalent to "real homotopy" which would suggest that the cofibrations should correspond very roughly to fibrations of spaces (judging from your example this looks like the notion you are searching for). Then the proper notion would seem to be a pseudofree extension algebra (i.e., the extension algebra forgetting the differential) should be a polynomial algebra over the base. In that case the map $\Omega^\bullet(X)\rightarrow\Omega^\bullet(X\times\mathbb R^n)$ is not a cofibration. I find it difficult to imagine an interesting model structure on commutative dga's which would make it a cofibration. 

