  In this article, we investigate the orbit configuration spaces of some
equivariant closed manifolds over simple convex polytopes in toric topology,
such as small covers, quasi-toric manifolds and (real) moment-angle manifolds;
especially for the cases of small covers and quasi-toric manifolds. These kinds
of orbit configuration spaces are all non-free and noncompact, but still built
via simple convex polytopes. We obtain an explicit formula of Euler
characteristic for orbit configuration spaces of small covers and quasi-toric
manifolds in terms of the $h$-vector of a simple convex polytope. As a
by-product of our method, we also obtain a formula of Euler characteristic for
the classical configuration space, which generalizes the F\'elix-Thomas
formula. In addition, we also study the homotopy type of such orbit
configuration spaces. In particular, we determine an equivariant strong
deformation retract of the orbit configuration space of 2 distinct orbit-points
in a small cover or a quasi-toric manifold, which turns out that we are able to
further study the algebraic topology of such an orbit configuration space by
using the Mayer-Vietoris spectral sequence.
