We investigate for the first time, to the best of our knowledge, both theoretically and experimentally, how the phase noise of the radio frequency (RF) drive signal affects the phase noise of the individual tones of a Fabry-Perot (F-P) modulator-based optical frequency comb. We observe that the expected deleterious effect of the RF drive signal phase noise on the comb output is partially suppressed due to the filtering characteristics of the F-P cavity. We found that the cavity-induced phase noise suppression is strongest for high-order comb tones, e.g., reaching up to 40 dB for the 100th comb tone at high offset frequencies. The phase noise suppression becomes even stronger for low RF-drive powers, or when the seed laser does not resonate in the F-P cavity. For both cases, we observe up to a 10 dB increase in phase noise suppression. We also evaluate the timing jitter improvement obtained, thanks to the cavity-induced phase noise reduction. The timing jitter (integrated from 2.5 MHz to 2.5 GHz) decreased by a factor of 7 for the beat signal obtained between two comb tones that are 100 tones apart (in comparison with the timing jitter obtained in a cavity-less comb generator).