Seeded free-electron lasers (FELs) hold great promise for generating high brilliance radiation pulses with a narrow bandwidth, which typically requires an electron bunch with relatively uniform energy distribution. However, it has been pointed out that the beam energy curvature generated in the acceleration process may degrade the output radiation pulse quality of seeded FELs. In this paper, we studied the beam energy chirp effects in various seeded FEL configurations. The theoretical and simulation results show that the performance degradation of high gain harmonic generation scheme is proportional to the beam energy chirp, while the advanced seeding schemes, e.g. echo-enabled harmonic generation and phase merging enhanced harmonic generation, are capable of eliminating the effect of the electron beam energy chirp.