$\DeclareMathOperator\SL{SL}\newcommand\O{\mathcal O_F}\newcommand\P{\pi\mathcal O_F}\newcommand\Pi{\pi^{-1}\mathcal O_F}$I assume that the standard Iwahori is the group of matrices in $\SL_3(\O)$ that are upper triangular modulo $\pi$. Then $B \cup Bw_1 B$ is $K \mathrel{:=} \begin{pmatrix} \O & \O & \Pi \\ \P & \O & \O \\ \P & \P & \O \end{pmatrix} \cap \SL_3(F)$.
Indeed, it is clear that this is a subgroup, that it contains (hence is stable under left- and right-multiplication by) $B$, and that it contains $w_1$. Therefore, it contains $B \cup Bw_1 B$.
On the other hand, suppose $g = \begin{pmatrix} a & b & \pi^{-1} c \\ \pi d & e & f \\ \pi g & \pi h & i \end{pmatrix}$ belongs to $K$ (so that $a, \dotsc, i$ belong to $\O$). If $c$ belongs to $\P$, then $g$ belongs to $B$. Otherwise, $b \mathrel{:=} \begin{pmatrix} 1 \\ & 1 \\ \pi i/c && 1 \end{pmatrix}$ and $(b w_1)^{-1}g$ both belong to $B$, so $g$ belongs to $B w_1 B$.