Another situation in which the result is true is that of an Abelian category with enough injectives. Here I take "injective map" to mean monomorphism. In such setting, choose a monomorphism $m_i \colon C_i \to I$, where $I$ is injective. Since $I$ is injective and $C_i \to C_{i + 1}$ is a monomorphism, you can lift $m_i$ to a morphism $m_{i+1} \colon C_{i+1} \to I$, and so on. By putting together all $m_i$, you get a morphism $m \colon C \to I$ such that $m_i$ factors as $C_i \to C \to I$. Since $m_i$ is a monomorphism, the morphism $C_i\to C$ is a monomorphism as well.
I learned this nice argument in Neeman - A counterexample to a 1961 "theorem" in homological algebra, where he also construct an example of an Abelian category and a chain of monomorphism such that the direct limit is $0$ - showing that the analogue of this result is not always true!
