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, 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 the 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 set-up. 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^2/Hz), and are thereby not identifiable.