Suppose we have some homotopical setting in which we can speak of homotopy colimits. The setting I have in mind at the moment is that of a compactly generated triangulated category with a model, but we could look at any model category or some other homotopical context. Let $A_0 \to A_1 \to A_2 \to \cdots$ be a system of objects indexed by the natural numbers. My question is if there are theorems of the form "under such and such hypotheses, hocolim $(A_i)$ agrees with colim $(A_i)$".

Suppose that we are in one of the two settings:
Then for any inductive system of objects indexed by natural numbers in C, the image in D of their colimit in C coincides with their homotopy colimit in D. But one needs Ab5 for this to hold in the case 1, and if one replaces colimits with limits in either case 1 or 2, the assertion will no longer hold. The proof is that in both cases 1 and 2 for any inductive system $X_i$ in C indexed by the natural numbers the telescope sequence $0\to \bigoplus_i X_i\to \bigoplus_i X_i\to \varinjlim X_i\to 0$ is exact in C, hence the cone of the left arrow is quasiisomorphic to the right term. (In fact, this argument applies already in the coderived category of complexes or DGmodules, so the assertions are true for the coderived categories. This allows to replace a DGring with a CDGring.) 


The obvious answer is: if all the maps are cofibrations (this might be your definition of hocolim). Or, slightly more generally, if a cofinal subsystem consists of cofibrations. It's still not an if and only if  are you looking for something stronger than that? 


Not an answer, but a clarifying comment. Language gets a little tricky here. For the sake of any beginners, and to try to standardize terms, let me say this. Recall that a diagram $\cal I\to \cal M$ of shape $\cal I$ in a model category $\cal M$ has a hocolim, welldefined up to weak equivalence. Better, there is a functor "hocolim" from the diagram category $\cal M^{\cal I}$ to $\cal M$, welldefined up to natural weak equivalence. And that this respects weak equivalences, meaning that if a map $X\to Y$ of diagrams is an objectwise weak equivalence (meaning that $X(i)\to Y(i)$ is an equivalence for all $i$) then the resulting map from $hocolim X$ to $hocolim Y$ is an equivalence. So this yields a functor from $w^{1}(\cal M^{\cal I})$ to $w^{1}\cal M$ that can also be called hocolim. It is a common misunderstanding to think that there in fact a related functor from $(w^{1}\cal M)^{\cal I}$ to $w^{1}\cal M$, but in general it's not true. You can have two diagrams $\cal I\to \cal M$ such that they yield isomorphic diagrams in $w^{1}M$ but have inequivalent hocolims. In particular, hocolim is not colim in $w^{1}\cal M$. Diagrams in $w^{1}\cal M$ don't always have colimits. When a diagram in $\cal M$ is such that as a diagram in $w^{1}\cal M$ it has a colim, then the latter must be a retract of the hocolim. In the special case where $\cal I$ is the ordered set of natural numbers, I find that I don't have anything very definitive to say. Certainly there are many examples of spaces or spectra where (sequential) hocolim is not colim in the homotopy category. 

