Let $\mathcal{F}\_1, \mathcal{F}\_2$ be coherent sheaves over $\mathbb{P}^n\_{\mathbb{C}}$ for $n \ge 3$. Now, $\Gamma\_\*(\mathcal{O}\_{\mathbb{P}^n})=\mathbb{C}[X\_0,...X\_n]$. Denote by $U\_0$ the affine scheme $\{X\_0 \not=0 \}$. Denote by $D\_1$ (resp. $D\_2$) the support of $\mathcal{F}\_1$ (resp. $\mathcal{F}\_2$). Assume that $D\_i \cap U\_0$ is dense in $D\_i$, for $i=1, 2$. Let $\phi\_0 \in H^0(U\_0,\mathcal{H}om\_{\mathbb{P}^n}(\mathcal{F}\_1,\mathcal{F}\_2)|\_{U\_0})$ be a morphism. Does there exist a morphism $\phi \in H^0(\mathbb{P}^n,\mathcal{H}om\_{\mathbb{P}^n}(\mathcal{F}\_1,\mathcal{F}\_2))$ such that $\phi|\_{U\_0}=\phi\_0$?



Extend morphism between coherent sheaves in $\mathbb{P}^n$
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. 


In the example of your interest, $\mathscr F\_1=\mathscr I\_Z$ and $\mathscr F\_2=\mathscr O\_Z$, pretty much the same thing is happening. Since here $\mathscr F\_2$ is supported on $Z$, working on $\mathbb P^n$ is almost a red herring. Morphisms between these sheaves, as you remark in a comment, are nothing else but sections of the normal sheaf of $Z$. So in this case your question translates to asking whether a section of this sheaf on an open (in fact a concrete open) set extends to the entire $Z$. Since you are working in $\mathbb P^n$ you know that this sheaf is ample, but that still does not mean that sections from open affine sets extend to global sections. In fact, observe that this will never happen. The space of sections of a line bundle on an affine set tend to be huge. This one is likely infinite dimensional. In contrast, the space of global sections is always finite dimensional. I think based on this you will find plenty of counter-examples and will see that this extension property rarely happens and probably never with these particular open sets. 


On the other hand, 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 $U$.


