Let $X,Y$ be defined over the field $k$ and take $f$ to be the structure map to ${\rm Spec}\, k$. Then let $E\to G$ be a surjective morphism of sheaves that is not surjective on global sections, e.g., $$\mathcal O_{\mathbb P^1}(-1)\oplus \mathcal O_{\mathbb P^1}(-1)\to \mathcal O_{\mathbb P^1}.$$ Then $f_*$ is just $H^0$ and the desired statement is false.

Let $X,Y$ be defined over the field $k$ and take $f$ to be the structure map $f:X\to {\rm Spec}\, k$. Then let $E\to G$ be a surjective morphism of sheaves that is not surjective on global sections, e.g., $$E=\mathcal O_{\mathbb P^1}(-1)\oplus \mathcal O_{\mathbb P^1}(-1)\to G=\mathcal O_{\mathbb P^1}.$$ Then $f_*$ is just $H^0$ and the desired statement is false.

EDIT: (to have an example mapping to an arbitrary scheme) Consider the base change of $f$ via $Y\to {\rm Spec}\, k$: $$g: X\times_{{\rm Spec}\, k} Y \to Y.$$ and let $\mathcal E:=p^*E$ and $\mathcal G:=p^*G$ where $p:X\times_{{\rm Spec}\, k} Y \to X$ is the projection to $X$. Then $g_*\mathcal E\simeq H^0(X, E)\otimes_k \mathcal O_Y$ and $g_*\mathcal G\simeq H^0(X, G)\otimes_k \mathcal O_Y$, so again the desired statement is false.