Let $R$ be a commutative ring with unity. $r$ is a non zero non unit in $R$. Then $R[X]/(rX)$ is not flat as an $R$ algebra if $R$ is an integral domain. But what can we say if $R$ is not an integral domain ?
EDIT: Consider a field $k$ and $R=k \oplus k$ equipped with the natural ring structure. Let $r=(1,0)$. Then $R[X]/(rX) = k[X] \oplus k$ which is projective over $R$. Does this example work?
Well, I interpret that to mean that you're looking for a ring that isn't a domain and has a quotient by a principal ideal that is flat.
Yes, it's projective, at least because $k$ is projective as an $R$ module ( as a summand of $R$), and $k[X]$ is projective (as a direct summand of the free $R$ module $R[X]$), and so is their product $k\times k[X]$.