When a regular $F_{\sigma}$ set open?

Using the correct definition

A regular $F_{\sigma}$ set is a set $S \subset X$ such that S is a union of a sequence of closed sets $F_n$ whose interiors contain $S$ i.e. $S=\cup_{1}^{\infty} F_n= \cup_{1}^{\infty} {F_n}^{\circ}$.

Then it's obvious that a regular $F_\sigma$ set is in particular an $F_\sigma$ (as witnessed by $S=\bigcup_n F_n$) and open (because $S=\bigcup_n F_n^\circ$, a union of opens). Not all open $F_\sigma$ sets will be regular $F_\sigma$ in general but as noted in the comments, this is the case in any metrisable space $X$, using the sets $F_n = \{x \in O\mid d(x,O^\complement) \ge \frac1n\}$ for an open set $O$. Also in regular hereditarily Lindelöf spaces this will be the case.