Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?

Toen proved that $RHom$ provides the internal hom in the homotopy category of dg-categories. For what you want to be true, you need something more than this: you need to know that $RHom$ is actually the right-derived functor of the strict internal hom $DG$. Apparently, this was proved in arXiv:1202.3359 (Prop. 5.2).

On the other hand, if you are willing to work up to derived Morita equivalence instead of quasi-equivalence, then there is the paper [G. Tabuada, Homotopy theory of dg categories via localizing pairs and Drinfeld’s DG quotient, HHA, 12 (2010), no. 1, p. 187-219]. Tabuada constructs a model category of localization pairs, which is Quillen equivalent to the Morita model structure on dg-categories, and shows that it admits a right-derived internal hom functor which agrees with Toen's construction.