Categories whose coproducts are not disjoint

Perhaps this is a cheap example, but given a set $X$, its power set $2^X$ is in particular a poset by the inclusion relation, and is thus a category. In this case, the coproduct of $U,V\subseteq X$ is given by their union, which is disjoint only if $U$ and $V$ were already disjoint as sets since $\require{AMScd}$ \begin{CD} U\cap V @>>> U \\ @VVV @VVV \\ V @>>> U\cup V \end{CD} is a pullback square (and a pushout square).

Here's an example which is even a familiar category of algebraic objects: in commutative rings the coproduct is the tensor product $\otimes$, and it sometimes happens that the tensor product of two rings is the zero ring. In that case the pullback of the diagram you're looking at is just the product. So, for example, $\mathbb{F}_2$ and $\mathbb{F}_3$ don't have a disjoint coproduct.

(However, coproducts are disjoint in the opposite category of affine schemes!)