1. The equality holds if $G(t,x) = f(t) + g(x)$.

2. If $G(t,x) = f(t) g(x)$, then
$$E(\sup(G(t,X))=\sup f(t) E(g(X)1_{g(X)\geq0}) + \inf f(t) E(g(X)1_{g(X)<0})$$
and
\begin{equation}
\sup E(G(t,X)) =
\begin{cases}
E(g(X)) \sup f(t) & \text{ if }  E(g(X)) \geq 0 \newline
E(g(X)) \inf f(t) & \text{ if }  E(g(X)) < 0 \end{cases}
\end{equation}
So the equality holds if $g(X) \geq 0$ a.s. or $g(X) \leq 0$ a.s.