Let $ R$ be a cofibrant commutative $ S$ -algebra (in the sense of Elmendorf-Kriz-Mandell-May; they call them “$ q$ -cofibrant”) and $ A$ be a cofibrant commutative $ R$ -algebra.

Does $ A\wedge_R-:RMod→RMod$ preserve all weak equivalences?

This could be rephrased as “$ A$ is a flat $ R$ -module”. It is true that cofibrant $ R$ -modules are flat (EKMM III.3.8), however, the underlying $ R$ -modules of cofibrant commutative $ R$ -algebras are not generally cofibrant. Another positive result is that in the above hypotheses, $ A\wedge_R-$ preserves weak equivalences of cofibrant commutative $ R$ -algebras (EKMM VII.7.4).