My question is related to this question: When is the pushforward of a quasi-coherent sheaf quasi-coherent? Hartshorne proof

In Hartshorne page page 115 Proposition 5.8 it has been proved that if $X$ is noetherian or if $f: X\rightarrow Y$ is quasi-compact and separated then if $\mathcal{F}$ is a quasi-coherent sheaf on $X$, then $f_*\mathcal{F}$ is a quasi-coherent sheaf on $Y$.

Now let's consider an open immersion of schemes $i: U\rightarrow X$. In general $i$ is not quasi-compact. But is it still true that if $\mathcal{F}$ is a quasi-coherent sheaf on $U$, then $f_*\mathcal{F}$ is a quasi-coherent sheaf on $X$?