  With the expected large number of binary neutron star (BNS) observations
through gravitational waves (GWs), third-generation GW detectors, Cosmic
Explorer (CE) and Einstein Telescope (ET), will be able to constrain the tidal
deformability, and hence the equation of state (EoS) of neutron star (NS) with
exquisite precision. A subset of the detected BNS systems can retain residual
eccentricity in the detector frequency band. We study the systematic errors due
to unmodeled eccentricity in the tidal deformability measurement and its
implications for NS EoS and redshift measurement via the Love siren method. We
find that the systematic errors in the tidal deformability parameter exceed the
statistical errors at an eccentricity of $\sim 10^{-3}$ ($\sim 3\times
10^{-4}$) at $10$Hz reference GW frequency for CE (ET). We show that these
biases on tidal deformability parameter can significantly bias the NS EoS
inference. Furthermore, the error on tidal deformability propagates to the
source frame NS mass, which in turn biases the redshift inference. For CE, the
redshift inference is significantly biased at an eccentricity of $\sim 10^{-3}$
(at a reference frequency of $10$Hz). We also study the implications of biased
tidal deformability in testing the Kerr nature of black holes. Systematic error
on the tidal deformability parameter leads to a non-zero value of tidal
deformability for binary black holes, indicating a false deviation from the
Kerr nature. Finally, we show that including eccentricity in the waveform model
increases the statistical errors in tidal deformability measurement by a factor
of $\lesssim 2$. Our study, therefore, highlights the importance of using
accurate eccentric waveform models for GW parameter inference.
