We study the midinfrared plasmonic response in Bernal-stacked bilayer graphene. Unlike its monolayer counterpart, bilayer graphene accommodates optically active phonon modes and a resonant interband transition at infrared frequencies. They strongly modify the plasmonic properties of bilayer graphene, leading to Fano-type resonances, giant plasmonic enhancement of infrared phonon absorption, a narrow window of optical transparency, and a new plasmonic mode at higher energy than that of the classical plasmon.