It's true for the category of sets and various related categories, such as topological spaces. It's false for pointed sets. And, as Laurent Moret-Bailly has pointed out, it's false for the opposite of the category of sets.
When it's true of a particular category, it may be because for each $Y\to X$ the pullback functor $U\mapsto U\times_XY$ has a right adjoint -- in which case pullbacks preserve all colimits, not just coproducts.
EDIT Take the category of commutative semigroups. (An object is a set with a commutative and associative addition law.) Categorical product is the obvious thing. The initial object (empty set) is strict in your sense. The coproduct of $A$ and $B$ is the disjoint union of $A$, $B$, and $A\times B$ with the obvious addition law. Product is not distributive over coproduct.
