An exact and faithful functor is conservative? Is it true that an exact and faithful functor $F$ is conservative? Here, exact means that it commutes with finite limits and colimits, and conservative means that if $F(f)$ is an isomorphism, then $f$ was also an isomorphism.

More generally, here is the sort of thing that can go wrong. A faithful functor reflects epimorphisms and monomorphisms; it follows that if $F(f)$ is an isomorphism, then $f$ is an epimorphism and a monomorphism. But there are categories in which such a map is not an isomorphism, for example $\text{Top}$ as in MatheinBoulemenos' answer.

On the positive side, this shows that if $F$ is faithful and the domain has the property that any map which is an epimorphism and a monomorphism is an isomorphism (e.g. it's an abelian category) then $F$ is conservative with no exactness hypotheses.