As long as the complement of your open set is "big" as in "codimension $1$", there is no hope to do anything like this. Here is a concrete example on $\mathbb P^n$: Let $\mathscr L_1$ and $\mathscr L_2$ be two arbitrary line bundles such that there exists no morphism $\phi:\mathscr L_1\to \mathscr L_2$. On $\mathbb P^n$ any line bundle restricts to the trivial line bundle on your $U_0$ so there will be a morphism there. 

The point is that the morphism that you observe on $U_0$ may have poles along the complement if the complement is codimension $1$. This is exactly what happens in this example. 

If you assume that $\mathscr F_1$ and $\mathscr F_2$ are reflexive sheaves on a normal variety (e.g., locally free, but typically ideal sheaves and structure sheaves are not reflexive), and that you have a morphism one an open subset $U$ whose complement has at least codimension two, then the morphism extends; it is simply the push-forward of the morphism on $U$.