Let $X$ be a smooth, projective variety, $V_1, V_2$ smooth, closed subvarieties of the same dimension and $E$ a locally free sheaf on $X$. There exist natural morphisms $$r_1: H^i_{V_1}(E) \to H^i(E) \mbox{ and } r_2: H^i_{V_2}(E) \to H^i(E). $$ Suppose that $V_1 \cap V_2=\emptyset$. The question is, for which values of $i$ can we say that $\mbox{Im}(r_1) \cap \mbox{Im}(r_2)=0$?

For $i=0$ this holds essentially by the definition of $H^i_V$.

For $i>0$ it generally fails. Let $X$ be a curve and $V_i$ arbitrary, different points. Then the complements $X\setminus V_i$ are affine and hence both $r_1$ and $r_2$ are surjective.