The example of an exact functor that doesn't commute with colimits in abelian category It is mentioned here that a right exact funtor commute with colimits only under some good conditions. 
I wonder if there is a concrete counterexample of an exact functor which doesn't commute with colimits.

Consider the functor $\text{Hom}(M, -)$ where $M$ is, to keep things concrete, an $R$-module and Hom takes values in abelian groups. This functor is exact iff $M$ is projective but commutes with colimits iff $M$ is finitely generated projective. So a counterexample is given by any projective module which is not finitely generated, for example an infinitely generated free module $\bigoplus_{i=1}^{\infty} R$. 
