In general, this is certainly not true. Take for example a space $X$ with an action by a group $G$. As a group acts by isomorphisms, it acts in particular by cofibrations. But the map $X/G \to X_{hG}\simeq EG \times_G X$ from the orbits to the homotopy orbits is usually not an equivalence if the action is not free.

A sequential colimit of cofibrations is always a homotopy colimit if the first object is cofibrant. This follows for example by the general theory of Reedy model structures: In the Reedy model structure a sequential diagram as above is cofibrant. If your model category is left proper, you can indeed drop the assumption that the first object is cofibrant.

In general, if you have a direct diagram category $\mathcal{D}$ such that for every $X \in \mathcal{D}$ the category of $Y\neq X$ mapping to $X$ has a terminal object or is empty, then the colimit of every $\mathcal{D}$-shaped diagram of cofibrant objects and cofibrations is a homotopy colimit. This follows again by the Reedy model structure on $\mathcal{D}$-shaped diagrams (as the Latching objects are very easy to determine).