I'm hoping that I understand directed colimits and finite limits correctly. If I don't, I'll happily delete this answer.

Here's an example that works for several categories: for example, Hausdorff topological spaces and (smooth) manifolds. Actually, in each we take the slice category of objects over the real line. Consider the coequaliser diagram of the two obvious inclusions ℝ∖{0} → ℝ⨿ℝ; each object here is viewed as a manifold over ℝ in the obvious way. The coequaliser of this diagram in the category of manifolds over ℝ is ℝ (it ought to be ℝ with a double point at 0, but then we take the Hausdorffification of that to get the real line itself). Now consider the product *in the slice category* of this diagram with the manifold {0}, viewed as a manifold over ℝ with the obvious inclusion. The product with ℝ∖{0} is the empty set and the product with ℝ⨿ℝ is {0}⨿{0} so the coequaliser is just {0}⨿{0} again, but the product with ℝ is just {0}.

Note that this doesn't go away if you work with non-Hausdorff objects, so it would still work if we'd started with either topological spaces or non-Hausdorff manifolds. However, the difference isn't quite so stark for these so it's not so good an example. What you get in these cases is that the product with the coequaliser is {0}⨿{0} with the *indiscrete* topology and the coequaliser of the products is {0}⨿{0} with the *discrete* topology.

(Full disclosure: this example is based on one taken from the page on Frölicher spaces on the n-lab which was written by ... er ... me. There it illustrates the fact that the category of Frölicher spaces is not locally cartesian closed. Note that this is one of the requirements for a topos. The fact that this example works for *manifolds* shows that there is no topos in which the category of smooth manifolds embeds in such a way as to preserve both limits and colimits. But that's getting off-topic.)