Stimulated Raman scattering (SRS) enabled by the emerging gas-filled low-loss anti-resonant hollow-core fiber (ARHCF) technology opens up a competitive way towards the development of novel lasers in the molecular fingerprint region. In this article, the characteristics of noise and long-term stability of near- and mid-infrared (near-IR and mid-IR) gas-filled fiber Raman lasers have been investigated for the first time. The results reveal that an increase in Raman pulse energy is associated with a decrease in noise, and that the relative pulse peak intensity noise (RIN) is always lower than the relative pulse energy noise (REN). We also demonstrate that long-term drift of the pulse energy and peak power are directly linked with the high amount of heat release during the Raman Stokes generation. The demonstrated noise and long-term stability performance provide necessary references for potential spectroscopic applications as well as further improvements of the emerging IR gas-filled ARHCF Raman laser technology.