Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories? Toen in The homotopy theory of dg-categories and derived Morita theory Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined
$$
RHom(C,D)=Int((C\otimes^{\mathbb{L}}D^{op})-Mod^{rqr}).
$$
where "rqr" stands for right quasi-representable and "Int" stands for the full subcategory consisting of cofibrant and fibrant objects.
On the other hand we can consider the dg-category of dg-functors between $C$ and $D$ and denote it by $DG(C,D)$. See Keller's On differential graded categories Section 2.3.
In general $DG(C,D)$ will be different from $RHom(C,D)$ (they are not even quasi-equivalent.) Nevertheless, is it true that their homotopy categories are equivalent? In other words, do we have
$$
H^0(DG(C,D))\cong H^0(RHom(C,D))?
$$
If not, do we have counter-examples?
 A: 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.
A: I think there is a counterexample. Take $C=\Delta^1$ the category freely generated over the diagram $0 \to 1$. Then, $RHom(\Delta^1,D)$ is the dg-category of homotopy coherent morphisms in $D$, described for example in "DG quotients of DG categories" by Drinfeld, whereas $DG(\Delta^1, D)$ is what Drinfeld calls (in the same article) the "stupid dg-category of morphisms". Even at the $H^0$ level, these categories are very different from each other: morphisms in $H^0(DG(\Delta^1, D))$ are homotopy classes of strictly commutative squares, whereas morphisms in $H^0(RHom(\Delta^1, D))$ are homotopy classes of homotopy coherent commutative squares. Actually, you may view $H^0(RHom(C,D))$ in general as the localisation of $H^0(DG(C,D))$ along natural transformations which are levelwise homotopy invertible (see Keller, "On differential graded categories").
Also, objects of $H^0(DG(C,D))$ are strict dg-functors, whereas objects of $H^0(RHom(C,D))$ are quasi-functors. They essentially coincide if $C$ is cofibrant; on the other hand, if $C$ is not cofibrant, then you have to choose a cofibrant replacement $C'$, and I actually don't know if there is an equivalence $H^0(DG(C,D)) \cong H^0(DG(C',D))$; I suspect that this is not the case, in general.
