As has already been said, the homotopy category does not admit filtered colimits in general, but it’s much worse than that. Even colimits in an $\infty$-category which don’t give rise to colimits in the homotopy category sometimes do give rise to *weak* colimits. (A weak colimit cocone gives the existence, but not the uniqueness, of the factorizations a colimit cocone gives.) This is the case, for instance, with sequential colimits, as well as those along any free category, at least in spaces.

So one might ask whether at least every filtered colimit in $\mathcal S$ gives rise to a *weak* filtered colimit in $h\mathcal S$. Alas, this is still not true. In our paper kindly referenced by Tim, Christensen and I give an $\aleph_1$-indexed sequence of spaces whose homotopy colimit is not a weak colimit in the homotopy category, namely the sequence mapping a countable ordinal $\alpha$ to the wedge of $\alpha$ circles. The homotopy colimit is a wedge of $\aleph_1$ circles, and the problem is that a map out of *that* just requires too much coherence to be constructed out of a cocone over countable wedges in $h\mathcal S$. So there is not much hope for filtered colimits in $h\mathcal S$. I expect the same counterexample would work, though I have no idea how to make the argument, in higher homotopy categories $h_n\mathcal S$.

Regarding “minimal” or “distinguished” weak colimits, the general idea is that you want some weak colimits which are distinguished up to at least non-unique isomorphism, as occurs for cones in triangulated categories. Since “homotopy colimits” of sequences in triangulated categories with countable coproduct a are constructed out of those coproduct together with cones, they are also distinguished in this sense.

It is possible to get at the idea of minimal weak colimit of at least a filtered diagram in a category which may not be triangulated, but which has some set of objects detecting isomorphisms, by asking that $Hom(S,\mathrm{wcolim} D_i)\cong \mathrm{colim} Hom(S, D_i)$ for every $S$ in your isomorphism-detecting set. Such weak colimits are then indeed determined up to isomorphism, and they’re also nice because they see the objects $S$ as compact.

However, this is not to say that such distinguished weak colimits are common! Our diagram from above actually admits no weak colimit which views even $S^1$ as compact in this way. (Though note that some weak colimit always exists-homotopy pushouts give weak pushouts, coproducts exist, and then the usual construction applies.)

If your category actually has a set of compact generators in a model, as for $D(R)$, then a distinguished weak colimit must come from a homotopy colimit. Franke gives an argument, cited in our paper, that on these grounds distinguished weak colimits of uncountable chains should essentially never exist in $D(R)$. The problem is that there’s a spectral sequence converging to homs out of a homotopy colimit indexed by $J$ whose $E_2$ page involves the derived functors $R^n\mathrm{lim}^J$ for all $n$. These derived functors were shown by Osofsky to be non vanishing up through $n$ when $J=\aleph_n$, and the homotopy colimit is a weak colimit only if the spectral sequence collapses, so this probably shouldn’t happen. However Franke doesn’t give an argument that there couldn’t in principle be enough unlikely differentials to produce the collapse.

Christensen and I tried for a while to work with the analogous spectral sequence for spaces, but it seemed to require a proficiency with calculating higher derived limits unsupported by the literature-Osofsky gives a special example, and for all I can tell nobody else has ever calculated a derived limit over $\aleph_n$. So our approach turns out to be entirely different and doesn’t immediately apply to the stable case. Thus I think it’s unknown, though highly doubtful, whether $D(R)$ admits minimal filtered colimits in general.