In a category is the direct limit an exact functor? Let $\mathcal{C}$ be a category and suppose that $\mathcal{C}$ admits direct limits. Let $\mathcal{C}_I$ the category of directed systems of objects of $\mathcal{C}$ indexed by $I$. If $\mathcal{C}$ is an exact category is easy to see that $\mathcal{C}_I$ is also, but is the functor $\varinjlim:\mathcal{C}_I\rightarrow\mathcal{C}$ exact?

No: Note that by passing to ${\mathscr C}^{\text{op}}$ you can equivalently ask for an example of an exact category in which directed inverse limits are not exact, and such is already provided by the category ${\textsf{Ab}}$ of abelian groups. See here. (Concretely, ${\textsf{Ab}}^{\text{op}}$ is equivalent to the category of compact Haussdorff abelian topological groups by Pontryagin duality)
Finally, note that you'll always have such kind of deficiency on either the limit or colimit side: Any abelian category satisfying both Grothendieck's axioms AB5 and AB5* is trivial.
