Is projection of a closed set $F\subseteq X\times Y$ always closed?

This is not always true. A counterexample is $X=Y=\mathbb{R}$, $F=\{(x,y)\in\mathbb{R}^2:xy=1\}$.

As you have known, it is not always true. However, putting some conditions on the spaces, we have the following lemma:

If $Y$ is compact, then the projection $\pi_1: X \times Y \to X$ is a closed map.

It means that if $C$ is closed in $X \times Y$, then $\pi_1(C)$ is closed in $X$.

You can find the proof of the lemma here.