In general, we have the socalled projection formula: if $f:X\to Y$ is a morphism of ringed spaces, $\mathcal F$ an $\mathcal O_X$-module, and $E$ be a locally-free $\mathcal O_Y$ module of finite rank, then $f_* (\mathcal F \otimes f^{*} E) \simeq f_{*}\mathcal F \otimes E$.

Edit (following Will's [remark](https://mathoverflow.net/questions/114032/restriction-of-sheaf#comment292160_114035)): The projection formula yields in the case of an open immersion $i:U \subset X$ the following identity : $i_* i^* F \simeq i_*\mathcal O_U \otimes F$. Therefore, if $U$ has codimension at least 2 in $X$, then $i_* i^* F\simeq F$ by normality of $X$.