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 :

$0 \to j_{!} ( \mathcal{F}_{U} ) \to \mathcal{F} \to i_{*} ( \mathcal{F}_{P} ) \to 0$

$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.