Here is one type of example, basically inspired by the manifold-type examples but so general that they are not actually categories of manifolds. Let $B$ be any small category with finite coproducts, and let $C$ be the category of diagrams of shape
$$X \to Z \leftarrow Y$$
in $B$. Coproducts in $C$ are given pointwise by coproducts in $B$. Define
$$\partial(X \to Z \leftarrow Y) = (0 \to X + Y \leftarrow 0)$$
with the obvious extension to morphisms. It is quite clear that $\partial$ preserves coproducts and that $\partial^2 \cong 0$. Also there is a canonical natural transformation $i: \partial \to 1_C$, whose component at the object $X \to Z \leftarrow Y$ is the unique one where the arrow in the middle is the map $X + Y \to Z$ whose restrictions to $X$ and $Y$ are the given arrows $X \to Z$, $Y \to Z$ of the object.