Claim: If the simplicial nerve of $I$ has dimension $m$ and $A$ has global dimension $n$, then $\operatorname{Fun}(I, A)$ has global dimension (at most) $m+n$.
To prove it, you can use the standard simplicial resolution of a functor $F\colon I \to A$ by representable functors.
$$F(-)\leftarrow \bigoplus_{c_0\in Ob(I)} F(c_0)\times \operatorname{mor}_I(c_0,-) \Leftarrow \bigoplus_{c_0\to c_1} F(c_0)\times \operatorname{mor}_I(c_1,-)\cdots$$
Taking $\operatorname{nat}(-, G)$ we obtain a cosimplicial resolution of $\operatorname{nat}(F, G)$. The cosimplicial object has dimension $m$, and each term has homological dimension $n$. It follows that $\operatorname{Ext}^k(F, G)=0$ for $k>m+n$.
