Cov. right-exact additive functors that don't commute with direct sums?

Here is a specific example, though it admits obvious generalizations. Let $R=S=\mathbb{Z}$, and consider the functor from abelian groups to abelian groups defined by $$T(X) = \mathrm{Ext}^1_{\mathbb{Z}}(\mathbb{Q}_p/\mathbb{Z}_p,X),$$ where $p$ is a prime. This is right-exact since $\mathrm{Ext}^2_{\mathbb{Z}}\equiv0$. It does not commute with direct sums, a fact which is clear when you observe that $T$ is also the $0$th left derived functor of $p$-completion, and $p$-completion does not preserve infinite sums, even when applied to free abelian groups.

Double dual (when $R=S$ is semisimple).

Or $X\mapsto \Pi_j (M_j\otimes_R X)$ where $\lbrace M_j\rbrace $ is an infinite collection of nontrivial bimodules.

What about representable functors? $\hom(M,-) : \mathrm{Mod}(R) \to \mathrm{Ab}$ is right exact iff $M$ is projective, and it preserves direct sums iff $M$ is even finitely generated projective. Of course this is almost a special case of the excellent answer by Tom Goodwillie, but I wanted to add this simple observation.

Remark: If $M$ is not projective, it is harder to characterize those modules $M$ such that $\hom(M,-)$ preserves direct sums, see this MO discussion and the associated nlab article.