If I make no mistake, one can construct a dg-lift as follows: The key point is that any sheaf of vectorspaces embedds canonical into a an injective sheaf of injective vectorspaces:
$$\mathcal F \rightarrow \prod_{x\in X} {i_x}_* {i_x}^* \cal F$$ By standard constructions this allows to construct natural injective resolutions of sheaves and even of bounded below complexes of sheaves. One can check that his actually yields a canonical dg-functor from the category of bounded below complexes of sheaves, to bounded below complexes of injective sheaves
$$I: C^+(X)\rightarrow C^+(\mathbb C-inj)$$
which maps each complex to a quasi-isomorphic complex of injectives.
Now let $\tilde \Omega$ be a finite flat resolution of the top forms. For example the usual $\mathcal D_{X^{an}}$ valued differential forms will do. We can now define $$\tilde{DR}:C^+(\mathcal D_X-inj) \rightarrow C^+(\mathbb C-inj)$$
from the dg-category of bounded below complexes of injective $\mathcal D_X$-modules with to the dg-category of bounded below complexes of injective sheaves by the formula:
$$\tilde{DR}(\mathcal M):=I(\tilde{\Omega}\otimes_{\mathcal D_{X^{an}} \mathcal M^{an}})$$
It is clear by construction that $\tilde{DR}$ induces the usual $DR$ on homotopy categories, hence $\tilde{DR}$ actually restricts to a dg-equivalence in the sense of Chris Brav's answer:
$$\tilde{DR}:C^b_{rh}(\mathcal D_X-inj) \rightarrow C^b_c(\mathbb C-inj)$$
from the dg-category of finte complexes of injective $\mathcal D_X$-modules with regular holonomic cohomology to the dg-category of complexes of injective sheaves with bounded constructible cohomology.
In fact there are functorial injective embeddings in many abelian categories and by the same recipe this should allow to construct dg-lifts of many functors. For example the duality functor, the solution functor etc.
