We analyze the impact of a proposed tidal instability coupling $p$-modes and $g$-modes within neutron stars on GW170817. This non-resonant instability transfers energy from the orbit of the binary to internal modes of the stars, accelerating the gravitational-wave driven inspiral. We model the impact of this instability on the phasing of the gravitational wave signal using three parameters per star: an overall amplitude, a saturation frequency, and a spectral index. Incorporating these additional parameters, we compute the Bayes Factor ($ln B^{pg}_{!pg}$) comparing our $p$-$g$ model to a standard one. We find that the observed signal is consistent with waveform models that neglect $p$-$g$ effects, with $ln B^{pg}_{!pg} = 0.03^{+0.70}_{-0.58}$ (maximum a posteriori and 90% credible region). By injecting simulated signals that do not include $p$-$g$ effects and recovering them with the $p$-$g$ model, we show that there is a $simeq 50%$ probability of obtaining similar $ln B^{pg}_{!pg}$ even when $p$-$g$ effects are absent. We find that the $p$-$g$ amplitude for 1.4 $M_odot$ neutron stars is constrained to $lesssim text{few}times10^{-7}$, with maxima a posteriori near $sim 10^{-7}$ and $p$-$g$ saturation frequency $sim 70, mathrm{Hz}$. This suggests that there are less than a few hundred excited modes, assuming they all saturate by wave breaking. For comparison, theoretical upper bounds suggest a $p$-$g$ amplitude $lesssim 10^{-6}$ and $lesssim 10^{3}$ modes saturating by wave breaking. Thus, the measured constraints only rule out extreme values of the $p$-$g$ parameters. They also imply that the instability dissipates $lesssim 10^{51}, mathrm{ergs}$ over the entire inspiral, i.e., less than a few percent of the energy radiated as gravitational waves.
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