  Dissipative tidal interactions can be used to probe the out-of-equilibrium
physics of neutron stars using gravitational wave observations. In this paper,
we present the first post-Newtonian (PN) corrections to the orbital dynamics of
a binary system containing objects whose tidal interactions have a dissipative
contribution. We derive the 1PN-accurate equations of motion in the
center-of-mass frame and a generalized energy-balance law that is valid for
dissipative tidal interactions. We show how mass and energy loss due to the
absorption of orbital energy change the orbital dynamics and derive the
next-to-leading order correction to the gravitational wave phase of a binary
system in a quasi-circular orbit containing initially non-spinning components.
We then use this waveform model to constrain, for the first time, the
individual dissipative tidal deformabilities of each of the binary components
that generated the GW170817 event using real data. We find that the GW170817
data requires $\Xi_{1} \lesssim 1121$ and $\Xi_{2} \lesssim 1692$ at 90\%
confidence, where $\Xi_{1,2}$ are the individual tidal deformabilities of the
primary and secondary binary components that produced the GW170817 event.
