  Rota-Baxter operators, $\mathcal{O}$-operators on Lie algebras and their
interconnected pre-Lie and post-Lie algebras are important algebraic structures
with applications in mathematical physics. This paper introduces the notions of
a homotopy Rota-Baxter operator and a homotopy $\mathcal{O}$-operator on a
symmetric graded Lie algebra. Their characterization by Maurer-Cartan elements
of suitable differential graded Lie algebras is provided. Through the action of
a homotopy $\mathcal{O}$-operator on a symmetric graded Lie algebra, we arrive
at the notion of an operator homotopy post-Lie algebra, together with its
characterization in terms of Maurer-Cartan elements. A cohomology theory of
post-Lie algebras is established, with an application to 2-term skeletal
operator homotopy post-Lie algebras.
