When is the Zariski topology $T_2$?

The Zariski topology on $F^n$ is never Hausdorff if $n>0$ and $F$ is infinite. Indeed, you can reduce to the case $n=1$ by noting that the subspace topology on $F\times \{0\}^{n-1}\subseteq F^n$ coincides with the Zariski topology on $F$ (identifying $F$ and $F\times \{0\}^{n-1}$ in the obvious way): given a polynomial $f(x_1,x_2,\dots,x_n)$, the subset of $F\times \{0\}^{n-1}$ on which it vanishes is just the subset of $F$ where the single-variable polynomial $f(x,0,\dots,0)$ vanishes. The Zariski topology on $F$ is just the cofinite topology since any nonzero single-variable polynomial can only vanish at finitely many points, and in particular is not Hausdorff. Since a subspace of a Hausdorff space is Hausdorff, this means $F^n$ cannot be Hausdorff.