A heavy impurity moving in a superfluid has remarkable properties. The medium dresses the impurity into a polaronic quasiparticle, and at strong binding the polaron can substantially deform the surrounding medium. When there are two or more polarons, fluctuations of the medium induce a Casimir interaction between polarons. We investigate this induced interaction using the Gross-Pitaevskii equation and effective field theory (EFT) and find regimes of scale invariance, Yukawa and power-law scaling.