A good example of the situation you are thinking about is the category of filtered modules (over some ring). As Yemon Choi notes in a comment, Banach spaces give another example (and in general, filtrations behave very similarly to topologies, and are a closely related notion); but (at least for me) it is a bit easier to write down filtered modules concretely.

Suppose that you have two exact sequences in the category of filtered modules (exact in the strong sense that the filtrations on the outer two terms are induced by the filtration on the middle term, which is to say that these morphisms satisfy the condition that coimage = image) and a morphism between them; the snake lemma will then give a long sequence which is exact as a sequence of morphisms of modules, but which is no longer exact in the strong sense that its morphisms need no longer be strict: i.e. they need not satisfy the condition that
coimage = image.

A trivial example is given by considering any morphism
$$( 0 \to A \to B \to C \to 0) \longrightarrow (0 \to A' \to B'\to C' \to 0)$$
of exact sequences of non-trivial $R$-modules, and then declaring $A$, $B$, and $C$ to be filtered by having $F^0A = A$ and $F^1A = 0$, and similarly with $B$ and $C$,
while equipping $A'$, $B'$, and $C'$ with filtrations such that $F^1A' = A'$ and similarly with $B'$ and $C'$.

The boundary map in the snake lemma from $ker(C \to C')$ to $coker(A\to A')$ now will not be strict (because $F^1$ on the kernel is $0$, while $F^1$ on the cokernel is everything).

One can nevertheless work out homological algebra in this context (e.g. one can form the "filtered derived category", which is the derived category of filtered modules) but one has to take care with the details, because of the phenomenon described above. See Laumon's paper "Sur la categorie derivees des $\mathcal D$-modules filtres" (Lecture Notes in Math 1016) for a careful development of the theory.