  Let $A$ be a commutative noetherian ring, let $\mathfrak{a}\subseteq A$ be an
ideal, and let $I$ be an injective $A$-module. A basic result in the structure
theory of injective modules states that the $A$-module
$\Gamma_{\mathfrak{a}}(I)$ consisting of $\mathfrak{a}$-torsion elements is
also an injective $A$-module. Recently, de Jong proved a dual result: If $F$ is
a flat $A$-module, then the $\mathfrak{a}$-adic completion of $F$ is also a
flat $A$-module. In this paper we generalize these facts to commutative
noetherian DG-rings: let $A$ be a commutative non-positive DG-ring such that
$\mathrm{H}^0(A)$ is a noetherian ring, and for each $i<0$, the
$\mathrm{H}^0(A)$-module $\mathrm{H}^i(A)$ is finitely generated. Given an
ideal $\bar{\mathfrak{a}} \subseteq \mathrm{H}^0(A)$, we show that the local
cohomology functor $\mathrm{R}\Gamma_{\bar{\mathfrak{a}}}$ associated to
$\bar{\mathfrak{a}}$ does not increase injective dimension. Dually, the derived
$\bar{\mathfrak{a}}$-adic completion functor
$\mathrm{L}\Lambda_{\bar{\mathfrak{a}}}$ does not increase flat dimension.
