  We use the path-integral formula and investigate some dynamical quantum
effects induced by the inspiraling lateral corrugation of orbital plane in
gravitationally bound neutron star (NS) binaries, with orbital separation of
$10^9$ m. Based on Dewitt's approach, we calculate the gravitational Casimir
energy cost of the binary plane, which consists of statically gravitational
effects and deformation-induced effects. It is found that the static effects
include a term coming from the self-gravity of the orbital plane and the
contribution of Newtonian gravitational potential of the binary system. While
the deformation-induced effect also results from two parts, i.e. the
instability of orbital binding energy, scaling as $\frac{1}{(R-r)^2}$, and the
dynamically Casimir energy cost of the orbital binding energy, decaying as
$\frac{1}{(R-r)^4}$. The dynamically gravitational Casimir phenomena and the
corresponding energy cost modify the spiral-in orbital motion of the binary and
thus the frequency of released gravitational waves (GWs). We consider the
mechanical response of two NS components and qualitatively study the
corrections to the orbital motion of the system and the GW frequencies. It is
found that the dynamical Casimir effects exert a dissipative force on the
binary plane, depending on the frequency of GWs. The resultant dissipation may
enhance with the decaying separation and increasing GW frequencies, which
subsequently accelerates the orbital decay of the binary. However, the
dissipation rate just has an order of $10^{-70}$ eV/s. So the corrections to
the dynamics of NS binaries are very marginal, by considering the wide
separation, the cosmological coalescence time, and low-frequency GWs of the
system.
