We evaluate the measurement errors induced by various deleterious effects in an optical passive ring-resonator gyro (OPRG) with a hollow-core photonic bandgap fiber (HC-PBF) sensing coil. The uncertainties in measuring rotation rate due to Kerr, Shupe, and Faraday effects are found to be reduced respectively by 2~3, 1, and 1~2 orders of magnitude as compared with an OPRG with a conventional single mode fiber (SMF) sensing coil of similar parameters. The errors due to shot and coherent backscatter noises are larger for the OPRG made of the current state-of-the-art HC-PBF than for the OPRG with a conventional SMF coil, but are expected to reduce in future with improved fiber manufacture technologies.