Here is an extended comment regarding Charles answer, including some more references.

In [this paper][1], 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][2], 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. 

[1]:http://arxiv.org/abs/1507.01564
[2]:http://arxiv.org/abs/1311.4128