Let $X \subseteq \mathbb P^n$ denote a projective variety. When $X$ is a Hausdorff space (we're talking about $X$ as about a topological space with Zariski topology)?
Almost never, I believe: only if $X$ is finite.
I need this fact to finish a simple proof but I haven't any ideas on how to prove it.
Irreducible varieties $X$ always have generic points: i.e. a point $x_0\in X$ such that $\{x_0\}$ is dense in $X$. If $X$ was Hausdorff, then $\{x_0\}$ would also be closed, so $X=\{x_0\}$ is a singleton.
If $X$ is not irreducible, apply this to each irreducible component, so that $X$ is finite and discrete.