Consider $X = \mathbb{P}^{1}$, $U =\mathbb{A}^{1}$ and j the inclusion of a affine chart in the projective line. The complement of the affine chart is a point P, let i be the inclusion of this point in the projective line. Consider the sheaf $\mathcal{F} = \mathcal{O}(-2)$ on the projective line. One has $dim H^{1}( \mathbb{P}^{1}, \mathcal{F} ) =1$. One has an exact sequence :
$0 \longrightarrow j_{!}(\mathcal{F}|_{U}) \longrightarrow \mathcal{F} i_{*}(\mathcal{F}|_{P}) \longrightarrow 0$
$i_{*}(\mathcal{F}|_{P})$ is a skyscraper sheaf over P so its $H^{1}$ is 0. $H^{1}$ of $\mathcal{F}$ is not 0. By long exact sequence in cohomology, we have $H^{1}$ of $j_{!}(\mathcal{F}|_{U})$ is not 0.
But on the other hand $\mathcal{F}|_{U}$ is a trivial sheaf over a affine scheme so its $H^{1}$ is 0.
This example seems to show that the expected relation is not true.