I suspect that Graham Denham's argument only shows that the weaker condition $\mathrm{Tor}_1(M,\kappa(P))=0$ is open, where $\kappa(P)$ is the residue field of $P$. With the problem as stated, here is a counterexample: for a field $k$, take $R=k[x,y]$ and $M=R/(x,y)$. Then it is easy to check that $\mathrm{supp}_{fl}(M)$ consists of:

$\bullet$ the generic point of $\mathrm{Spec}(R)$,

$\bullet$ all closed points except the origin $(x,y)$, and

$\bullet$ all principal primes $(f)$ where $f$ is irreducible and not in $(x,y)$, i.e. all generic points of curves not containing the origin.

This is not open in $\mathrm{Spec}(R)$.