  Gravitational waves (GWs) from inspiralling neutron stars afford us a unique
opportunity to infer the as-of-yet unknown equation of state of cold hadronic
matter at supranuclear densities. During the inspiral, the dominant matter
effects are arise due to the star's response to their companion's tidal field,
leaving a characteristic imprint in the emitted GW signal. This unique
signature allows us to constrain the cold neutron star equation of state. At GW
frequencies above $\gtrsim 800$Hz, however, subdominant tidal effects known as
dynamical tides become important. In this letter, we demonstrate that
neglecting dynamical tidal effects associated with the fundamental ($f$-) mode
leads to large systematic biases in the measured tidal deformability of the
stars and hence in the inferred neutron star equation of state. Importantly, we
find that $f$-mode dynamical tides will already be relevant for Advanced LIGO's
and Virgo's fifth observing run ($\sim 2025$) -- neglecting dynamical tides can
lead to errors on the neutron radius of $\mathcal{O}(1{\rm km})$, with dramatic
implications for the measurement of the equation of state. Our results
demonstrate that the accurate modelling of subdominant tidal effects beyond the
adiabatic limit will be crucial to perform accurate measurements of the neutron
star equation of state in upcoming GW observations.
