Let $X$ be a smooth $\mathbf{C}$-scheme of finite type. In section 7.2 of their preprint "Quantization of Hitchin's Integrable System and Hecke Eigensheaves", Beilinson and Drinfeld define an adjunction
$$\mathcal{D}:Z^0(\mathbf{dgMod}_{\mathrm{qcoh}}(\Omega^{\cdot}_{X/\mathbf{C}}))\leftrightarrows \mathbf{Cplx}(\mathbf{Mod}^{\mathrm{r}}_{\mathrm{qcoh}}(\mathcal{D}_X))):\Omega$$
between the category of (closed morphisms of) quasi-coherent dg-modules over the de Rham algebra of $X/\mathbf{C}$ and that of complexes of quasi-coherent right $\mathcal{D}_X$-modules. Their aim is to obtain a derived equivalence between a certain localization of this category of dg-modules over $\Omega^{\cdot}_{X/\mathbf{C}}$ and the usual derived category of the abelian category $\mathbf{Mod}^{\mathrm{r}}_{\mathrm{qcoh}}(\mathcal{D}_X)$. The functor $\mathcal{D}$ does not send all quasi-isomorphisms to quasi-isomorphisms and Beilinson and Drinfeld's solution is to invert the morphisms of dg-modules that are sent to quasi-isomorphisms by $\mathcal{D}$, which they call $\mathcal{D}$-quasi-isomorphisms. In 7.2.6(iii) (and also in 2.1.10 of their "Chiral algebras"), Beilinson and Drinfeld assert that any quasi-isomorphism of bounded $\mathcal{O}_X$-coherent dg-$\Omega^{\cdot}_{X/\mathbf{C}}$-modules is a $\mathcal{D}$-quasi-isomorphism (the converse is true without the finiteness hypothesis). In light of Kapranov's paper ("On dg-modules over the de Rham complex and the vanishing cycles functor"), this sounds reasonable.
On the other hand, as suggested by 6.23 in these lecture notes, if we take $X=\operatorname{Spec}(\mathbf{C}[t])$ and consider the left $\mathcal{D}_X$-module $\mathcal{O}_X\operatorname{e}^t$ given by the free $\mathcal{O}_X$-module of rank $1$ generated by $\operatorname{e}^t$, i.e. the integrable connection $f\mapsto (f+f')\mathrm{d}t:\mathbf{C}[t]\to\mathbf{C}[t]\mathrm{d}t$, then its de Rham complex appears to be acyclic, $\mathcal{O}_X$-coherent and bounded. I think this means that the image of the corresponding right $\mathcal{D}_X$-module under the functor $\Omega$ is acyclic, bounded and $\mathcal{O}_X$-coherent, hence $\mathcal{D}$-acyclic. As the adjuction morphism $\mathcal{D}\Omega\to\operatorname{id}$ is a quasi-isomorphism by [BD, 7.2.4], the right $\mathcal{D}_X$-module corresponding to $\mathcal{O}_X\operatorname{e}^t$ is zero, which sounds absurd.
Question: Have I made a stupid error somewhere and, if not, how does one reconcile this example with Beilinson-Drinfeld's description of $\mathcal{D}$-quasi-isomorphisms for $\mathcal{O}_X$-coherent $\Omega^{\cdot}_{X/\mathbf{C}}$-modules?
Best Answer
Let $\mathcal M$ be a left $\mathcal D$-module over a smooth curve $X$. Then the Koszul duality functor assigns to $\mathcal M$ the DG-module $\Omega_X\otimes_{\mathcal O_X}\mathcal M$ over the de Rham complex $\Omega_X$. Viewed as a complex of sheaves, $\Omega_X\otimes_{\mathcal O_X}\mathcal M$ is a two-term complex whose terms are quasi-coherent $\mathcal O_X$-modules, but the differential is not $\mathcal O_X$-linear. For this reason, even when $X$ is affine, one has to distinguish between the acyclicity of the complex of global sections of $\Omega_X\otimes_{\mathcal O_X}\mathcal M$ and the acyclicity of this complex of sheaves itself.
Explicitly, let $x$ be a global coordinate on $X$ (assuming that one exists). Then the differential in the complex $\Omega_X\otimes_{\mathcal O_X}\mathcal M = (\mathcal M\to \Omega_X^1\otimes_{\mathcal O_X}\mathcal M)$ has the form $m\mapsto dx\otimes \partial/\partial x(m)$. The question about acyclicity of this complex of sheaves is, therefore, the question about injectivity and surjectivity of the operator $\partial/\partial x$ acting in the sections of $\mathcal M$.
In the case at hand, we have $\mathcal M=\mathcal O_X e^x$. So global sections of $\mathcal M$ over $\operatorname{Spec} \mathbb C[x]$ are expressions of the form $p(x)e^x$, where $p(x)$ is a polynomial in $x$. Hence one can easily see that the complex of global sections of $\Omega_X\otimes_{\mathcal O_X}\mathcal M$ over $\operatorname{Spec} \mathbb C[x]$ is acyclic.
The complexes of sections of $\operatorname{Spec} \mathbb C[x]$ over Zariski open subsets of $\operatorname{Spec} \mathbb C[x]$ are not acyclic, however. It suffices to consider sections over $\operatorname{Spec} \mathbb C[x,x^{-1}]$. As is well known, the function $x^{-1}e^x$ does not lie in the image of $\partial/\partial x$ acting in the space of Laureant polynomials in $x$ multiplied with $e^x$. Therefore, the two-term complex of sheaves $\Omega_X\otimes_{\mathcal O_X}\mathcal M$ is not acyclic. Its differential is an injective, but not surjective morphism of sheaves of $\mathbb C$-vector spaces over $\operatorname{Spec} \mathbb C[x]$.
On the other hand, one can consider the complex of sheaves of analytic forms $\Omega_X^{an}\otimes_{\mathcal O_X}\mathcal M$ in the analytic topology of the set of closed points of $\operatorname{Spec} \mathbb C[x]$. Then every function from $\mathcal O_X^{an}e^x$ will have a primitive analytic function locally in the analytic topology. So the differential in the two-term complex of sheaves $\Omega_X^{an}\otimes_{\mathcal O_X}\mathcal M$ is now surjective. However, it is no longer injective, as the constant functions are sections of $\mathcal O_X^{an}e^x$, the function $e^{-x}$ being analytic. This is what Kapranov is doing in his paper.
Having, as I hope, answered your question, let me now point out, as a side note, that what I would consider a superior alternative of the $\mathcal D{-}\Omega$ duality theory of the Beilinson--Drinfeld preprint can be found in Appendix B to my AMS Memoir "Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence", http://arxiv.org/abs/0905.2621 .
The point is that what you call "$\mathcal D$-quasi-isomorphisms" are defined in my paper in terms intrinsic to DG-modules over $\Omega$ (without any reference to differential operators). The corresponding localization is called the coderived category of quasi-coherent DG-modules over $\Omega_X$, and subsequently it is proven to be equivalent to the derived category of quasi-coherent $\mathcal D_X$-modules (for a smooth variety $X$ over any field, $\mathcal D_X$ denoting the crystalline differential operators in the case when the characteristic is finite).