Extension by zero not Quasi-coherent.

Consider the short exact sequence $$0 \longrightarrow i_!(\mathcal{O}_X\rvert_U) \longrightarrow \mathcal{O}_X \longrightarrow j_*(\mathcal{O}_X\rvert_Z) \longrightarrow 0$$ where $Z = X \setminus U$, and $j \colon Z \hookrightarrow X$ is the inclusion map, from Exercise II.1.19(c). Taking sections on $V$, we have the exact sequence $$0 \longrightarrow \Gamma(V,i_!(\mathcal{O}_X\rvert_U)) \longrightarrow \Gamma(V,\mathcal{O}_X) \longrightarrow \Gamma(V \cap Z,\mathcal{O}_X\rvert_Z)$$ But the map $\Gamma(V,\mathcal{O}_X) \to \Gamma(V \cap Z,\mathcal{O}_X\rvert_Z)$ is injective, since the composition $$\Gamma(V,\mathcal{O}_X) \longrightarrow \Gamma(V \cap Z,\mathcal{O}_X\rvert_Z) \longrightarrow \mathcal{O}_{V,z}$$ where $z \in V \cap Z$ is injective (see Prop. 3.29 in Görtz/Wedhorn). Thus, $\Gamma(V,i_!(\mathcal{O}_X\rvert_U)) = 0$ by exactness.

EDIT: Thank you to MooS for the injectivity argument!