Commutativity of the square diagram coming from an adjoint triple

First, a counterexample when $G$ is not fully faithful: take $G$ to be the diagonal functor $\Delta : \mathbf{Set} \to \mathbf{Set}$; then its left adjoint is disjoint union and its right adjoint is cartesian product, and the two canonical maps $(X \times Y) \amalg (X \times Y) \to (X \amalg Y) \times (X \amalg Y)$ are different in general.

On the other hand, it is true when $G$ is fully faithful. In fact, it is enough to assume that $G \bar{\eta}$ is a natural isomorphism. Then, $(G \bar{\eta})^{-1} = \bar{\epsilon} G$, so $G \bar{\eta} \bullet \bar{\epsilon} G = \mathrm{id}_{G H G}$, and: \begin{align} H \eta \bullet \epsilon H & = \epsilon H G F \bullet F G H \eta \\ & = \epsilon H G F \bullet (F G \bar{\eta} F \bullet F \bar{\epsilon} G F) \bullet F G H \eta \\ & = (\bar{\eta} F \bullet \epsilon F) \bullet (F \eta \bullet F \bar{\epsilon}) \\ & = \bar{\eta} F \bullet F \bar{\epsilon} \end{align}

If $R \to S$ is a ring homomorphism, we have an adjoint triple $$- \otimes_R S \dashv U \dashv \hom_R(U(S),-),$$ where $U : \mathsf{Mod}(S) \to \mathsf{Mod}(R)$ is the forgetful functor. Your diagram evaluated at some $M \in \mathsf{Mod}(R)$ looks as follows (I will omit $U$ from the notation): $$\begin{array}{cc} \hom_R(S,M) \otimes_R S & \rightarrow & M \otimes_R S \\ \downarrow && \downarrow \\ \hom_R(S,M) & \rightarrow & \hom_R(S,M \otimes_R S)\end{array}$$ It maps $f \otimes s$ to $f(1) \otimes s \in M \otimes_R S$, which is mapped to $t \mapsto f(1) \otimes s t$ in $\hom_R(S,M \otimes_R S)$. In the other direction, it maps $f \otimes s$ to $t \mapsto f(st)$ in $\hom_R(S,M)$, which is then mapped to $t \mapsto f(st) \otimes 1$ in $\hom_R(S,M \otimes_R S)$. From this we see that the diagram isn't commutative in general.

Moreover, it is commutative for all $M$ iff $s \otimes 1 = 1 \otimes s$ for all $s \in S$, which is equivalent to $R \to S$ being an epimorphism, which is equivalent to the fully faithfulness of $U$. After seeing Zhen Lin's answer, this is no surprise.