In this letter, we demonstrate that the interplay between Raman pump relative intensity noise and cross-phase modulation leads to a relative phase noise (RPN) that brings non-negligible performance degradation to coherent optical orthogonal frequency-division multiplexing (CO-OFDM) transmission systems with co-pumped Raman amplification. By theoretical analysis and numerical simulation, we proved that RPN brings more system impairment in terms of Q-factor penalty than the single carrier system, and relatively larger walk-off between pump and signal helps to suppress the RPN induced impairment. A higher-order modulated signal is less tolerant to RPN than a lower-order signal. With the same spectral efficiency, the quadrature-amplitude modulation format shows better tolerance to RPN than phase-shift keying. The reported findings will be useful for the design and optimization of Raman amplified CO-OFDM multi-carrier transmission systems.