We present a comprehensive phase noise characterization of a mid-IR Cr:ZnS frequency comb. Despite their emergence as a platform for high-resolution dual-comb spectroscopy (DCS), detailed investigations into the phase noise of Cr:ZnS combs have been lacking. To address this, we use a recently proposed phase noise measurement technique that employs multi-heterodyne detection and subspace tracking. This allows for the measurement of common mode, repetition rate, and high-order phase noise terms and their corresponding scaling as a function of a comb-line number, using a single measurement setup. We demonstrate that the comb under test is dominated by the common mode phase noise, while all the other phase noise terms are below the measurement noise floor (∼-120 dB rad