We numerically investigate the influence of high-order dispersion (HOD) on temporal and spectral characteristics of microresonator-based optical frequency combs. Theoretical analysis based on the moment method associated with numerical simulations are utilized to study the comb evolution dynamics, showing that temporal shifts of steady-state intracavity solitons are induced by high-odd-order dispersion rather than high-even-order dispersion. The role of HOD on comb spectral envelopes is also elucidated through analyzing the intracavity dispersive wave generations. We further demonstrate that the spectral envelope of a broadband optical frequency comb can be engineered by using a cavity dispersion profile with multiple zero dispersion wavelengths.