We present new Atacama Large Millimeter Array observations of the molecular gas and far-infrared continuum around the brightest cluster galaxy (BCG) in the cool-core cluster MACS 1931.8-2635. Our observations reveal (1.9 ± 0.3) × 1010 M ⊙ of molecular gas, on par with the largest known reservoirs of cold gas in a cluster core. We detect CO(1‑0), CO(3‑2), and CO(4‑3) emission from both diffuse and compact molecular gas components that extend from the BCG center out to ∼30 kpc to the northwest, tracing the UV knots and Hα filaments observed by the Hubble Space Telescope. Due to the lack of morphological symmetry, we hypothesize that the ∼300 km s‑1 velocity of the CO in the tail is not due to concurrent uplift by active galactic nucleus (AGN) jets; rather, we may be observing the aftermath of a recent AGN outburst. The CO spectral line energy distribution suggests that molecular gas excitation is influenced by processes related to both star formation and recent AGN feedback. Continuum emission in Bands 6 and 7 arises from dust and is spatially coincident with young stars and nebular emission observed in the UV and optical. We constrain the temperature of several dust clumps to be ≲10 K, which is too cold to be directly interacting with the surrounding ∼4.8 keV intracluster medium (ICM). The cold dust population extends beyond the observed CO emission and must either be protected from interacting with the ICM or be surrounded by local volumes of ICM that are several keV colder than observed by Chandra.