The functor $F$ you are looking for can be described as follows. Given a model category $M$, with class of weak equivalences $W$, one may associate to it an $\infty$-category $M_\infty$, equipped with a map $M \to  M_\infty$, which is characterized by the following universal property: For every $\infty$-category $D$, the natural map
$$Fun(M_\infty,D) \to Fun(M,D)$$
is fully faithful, and its essential image is spanned by those functors $M \to D$ which send $W$ to equivalences. 

The $\infty$-category $M_\infty$ may be constructed follows: Construct the Hammock localization $L^H(M,W)$ of $M$ with respect to $W$, which is a simplicial category. The $\infty$-category $M_\infty$ can then be obtained by taking the coherent nerve of any fibrant model of $L^H(M,W)$ (with respect to the Bergner model structure).
We refer to

http://arxiv.org/abs/1311.4128

for the above universal property. 

Note that the above construction works for any *relative category*, that is, a category equipped with a subcategory of weak equivalences containing all the objects. Let us call this procedure $\infty$-localization.  

Morphisms of model categories are given by Quillen pairs:
$$F:M\rightleftarrows N:R.$$
According to  Proposition 1.5.1 in the paper above, every Quillen pair as above induces an adjoint pair of $\infty$-categories:
$$F_\infty:M_\infty\rightleftarrows N_\infty:R_\infty.$$
It can be shown that a Quillen equivalence goes to an equivalence of $\infty$-categories by this construction.

So consider the category of model categories and left Quillen functors between them. This is actually a relative category, where the weak equivalences are the Quillen equivalences. The above construction defines a relative functor from this relative category to the relative category of $\infty$-categories (weak Kan-complexes and simplicial maps between them) with the weak equivalences being the equivalences of $\infty$-categories. This relative functor should be a derived  fully faithful embedding of the $\infty$-category of model categories (in the sense of the $\infty$-localization above) into the $\infty$-category of $\infty$-categories and *left adjoints* between them. (Perhaps there are some delicate technical issues here that I am suppressing.)