Here is an extended comment regarding Charles answer, including some more references.
In this paper, Theorem 2.5.9, it is shown that every model category (not necessarily a combinatorial one) has all limits and colimits. However, it is not hard to find examples of model categories whose underlying $\infty$-categories are neither presentable nor co-presentable. For instance, Isaksen's strict model structure on pro-simplicial sets. It is shown in the paper mentioned above that the underlying $\infty$-category of this model category is the pro category of spaces considered in Lurie's "Higher Topos Theory" Definition 7.1.6.1. The pro category of a large cocomplete and finitely complete $\infty$-category is complete and cocomplete but neither presentable nor copresentable.
In this paper, Proposition 1.5.1, it is shown that any Quillen pair between model categories (not necessarily combinatorial ones) gives rise to an adjoint pair of $\infty$-categories.
It seems plausible to me that the (underlying $\infty$-category of the) relative category of model categories and left Quillen functors between them, with weak equivalences taken to be the Quillen equivalences is equivalent to the $\infty$-category of complete and cocomplete $\infty$-categories and left adjoints between them. Restricting to combinatorial model categories would correspond to restricting to presentable $\infty$-categories in the image. The latter statement is probably already proven, whereas I am pretty sure the former is not.
