There are really two separate things being asked. (1) When is the complex $F\otimes G$ exact? (2) If it is exact, when is $F\otimes G$ a minimal free resolution?

The first question is computed by Tor. Namely $F\otimes G$ is exact if and only if $\text{Tor}_i(S/I,S/J)=0$ for all $i>0$

I believe that the second question is easier. Since the differential $\partial$ on $F\otimes G$ is defined in terms of differentials on $F$ and $G$ (which were assumed to be minimal free resolutions), we see that $\partial (F\otimes G)_i$ belongs to the maximal ideal times $(F\otimes G)_{i-1}$. Thus, $F\otimes G$ is a minimal free resolution if and only if it is exact.

Of course, in your example where $S=k[x_1,\dots,x_n,y_1,\dots,y_m]$, and $I$ only involves $x$-variables and $J$ and only involves $y$-variables, then the higher Tor's vanish and thus $F\otimes G$ is a minimal free resolution.