Let $\mathcal{D}$ be an additive category, and $f$ a morphism that is both monic and epic. Must $f$ be an isomorphism? Basic theory of Abelian categories tells us that this is true if $\mathcal{D}$ is abelian. However, is this still true if we don't necessarily have kernels or cokernels? 
Tag 05R4 in the stacks project (Derived Categories, Lemma 5.3) seems to implicitly assume that this is true.

Nope. Topological abelian groups form an additive category, as can be seen directly or as a property of the category of abelian group objects in any category with finite products-a biproduct of $x,y$ in any category enriched over abelian groups is given by an object $z$ with morphisms $i_1,i_2:x,y\to z,p_1,p_2:z\to x,y$ such that $p_ji_j=1,p_0i_1=p_1i_0=0, i_1p_1+i_2p_2=1$, as follows from Yoneda and the same result for abelian groups, and if $C$ has finite products then we can construct such a diagram given abelian group objects $x,y$ with $z=x\times y$. 
Anyway, it's just as easy to get nontrivial monic epics in AbTopGp as in Top. Consider the discrete and the indiscrete topologies on any abelian group, for instance, using that discrete and indiscrete spaces are closed under (finite) products.
