  Self-gravitating bodies can have an arbitrarily complex shape, which implies
a much richer multipolar structure than that of a black hole in General
Relativity. With this motivation, we study the corrections to the dynamics of a
binary system due to generic, nonaxisymmetric mass quadrupole moments to
leading post-Newtonian (PN) order. Utilizing the method of osculating orbits
and a multiple scale analysis, we find analytic solutions to the precession and
orbital dynamics of a (generically eccentric) binary in terms of the
dimensionless modulus parameters $\epsilon_{m}$, corresponding to axial $m=1$
and polar $m=2$ corrections from oblateness/prolateness. The solutions to the
precession dynamics are exact for $0 \le \epsilon_{2} < 1$, and perturbative in
$\epsilon_{1} \ll 1$. We further compute the leading order corrections to the
gravitational wave amplitude and phase for a quasi-circular binary due to mass
quadrupole effects. Making use of the stationary phase approximation and
shifted uniform asymptotics (SUA), the corrections to the phase enter at
relative 2PN order, while the amplitude modulations enter at -0.5PN order with
a SUA amplitude correction at 3.25PN order, relative 2PN order to the leading
order SUA correction. By investigating the dephasing due to generic quadrupole
moments, we find that a phase difference $\gtrsim 0.1$~radians is achievable
for $\epsilon_{m} \gtrsim 10^{-3}$, which suggests that constraints with
current and future ground-based gravitational wave detectors are possible. Our
results can be implemented in parameter estimation studies to constrain generic
multipolar deformations of the Kerr geometry and of neutron stars.
