Let $R$ be an integral domain and let $M$ be a flat $R$-module. Then $M$ is torsion-free, so $am \neq 0$ for all $0 \neq a \in A$ and $0 \neq m \in M$. In particular, if $0 \neq a \in A$, then $aM \neq 0$. So, $Ann_R(M) = 0$.
But then this question implies that all finitely generated flat modules are already faithfully flat, so this seems to be wrong.
So, how can a torsion-free module have non-trivial annihilator? I must be misunderstanding something!
I also thought that when $M$ is torsion-free, then for any prime ideal $P$ of $R$ the localization map $M \rightarrow M_P$ is injective, so if $M$ is non-zero so is $M_P$ and therefore $Supp(M) = Spec(R)$, implying $Ann(M) = 0$ for $M$ finitely generated.
What is wrong here? This must be absolutely stupid.