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Gravitational-wave observations of binary neutron star systems can shed light on the currently unknown dense matter equation of state. The equation of state determines a large number of neutron star properties, such as tidal deformability, radius, and quadrupole moment, several of which directly affect the emitted gravitational-wave signals. To reduce the dimensionality when computing gravitational waves and when interpreting observational data, quasiuniversal relations are commonly employed to connect different neutron star properties. However, quasiuniversal relations are not exact, and their use may introduce uncertainty and bias. We explore the potential biases arising from different quasiuniversal relations in the third-generation era: (i) the Love-Q relation connecting the spin-induced quadrupole moment and the tidal deformability, (ii) the relation between the fundamental mode frequency and the tidal deformability, and (iii) the binary Love relation. We find that for the quadrupole relation biases are only present for rapidly rotating systems, for the binary-Love relation induces moderate biases only in the next-to-leading-order tidal parameters, which can, however, propagate into the inferred equation of state at low masses. Regarding fundamental mode frequencies, we find that the employed relation introduces only negligible biases, while waveform systematic effects can become comparatively large. Our results highlight that, while quasiuniversal relations remain a useful tool within gravitational-wave analyses, careful treatment is needed to avoid biases in equation-of-state measurements with next-generation detectors.