Example of a functor that doesn't reflect isomorphism

A very generic counterexample:

Take for $\mathcal{D}$ the category with one object $O$ and one morphism $f$. There is a functor $F$ from any category to $\mathcal{D}$ that sends every object to $O$ and any morphism to $f$.

Since $f$ is an isomorphism this generates a counterexample for every category which contains at least one morphism which is not an isomorphism.

Hint: look at any continuous bijection that fails to be a homeomorphism.

Just take the forgetful functor $F$ from $\mathit{Top}$ to $\mathit{Set}$. Then, take a bijection $f$ between two topological spaces which is not a homeomorphism. Of course, $Ff$ will be an isomorphism.