Let $X$ be a topological space and $U\subset X$ an open subset. Let's work in the category of sheaves of abelian groups on $X$. Consider the constant sheaf on $U$, $\mathbb{Z}_U$, given by $\mathbb{Z}_U(V)=\{\text{continuous maps }U\cap V\rightarrow \mathbb Z\}$, where $\mathbb Z$ is given the discrete topology. I've been struggling to derive from Yoneda's lemma the formula $\hom(\mathbb{Z}_U,F)=F(U)$. Is this a consequence of Yoneda? If so, how? If it doesn't follow from Yoneda, is it true at all? If not, how can one compute $\hom(\mathbb{Z}_U,\mathbb{Z}_V)?$

This is not true; for example, take $X = \mathbb R^2$, $U = \mathbb R^2 \smallsetminus \{(0,0)\}$. Then your $\mathbb Z_U$ coincides with $\mathbb Z_X$, and $Hom(\mathbb Z_U, F)$ is $F(X)$, not $F(U)$. For the formula to hold you have to take as $\mathbb Z_U$ the extension of the constant sheaf by $0$, which is a very different animal. 


See [Tamme, Introduction to étale cohomology], p. 31, Remark (2.1.3). 


Pierre Schapira has some very nice, and more elementary notes that could help you. 

