We study the effects of quasiparticle interactions in a quasi-two dimensional (quasi-2D), zero-temperature Bose-Einstein condensate of dipolar atoms, which can exhibit a roton-maxon feature in its quasiparticle spectrum. Our focus is the Beliaev damping process, in which a quasiparticle collides with the condensate and resonantly decays into a pair of quasiparticles. Remarkably, the rate for this process exhibits a highly non-trivial dependence on the quasiparticle momentum and the dipolar interaction strength. For weak interactions, the low energy phonons experience no damping, and the higher energy quasiparticles undergo anomalously weak damping. In contrast, the Beliaev damping rates become anomalously large for stronger dipolar interactions, as rotons become energetically accessible as final states. Further, we find a qualitative anisotropy in the damping rates when the dipoles are tilted off the axis of symmetry. Our study reveals the unconventional nature of Beliaev damping in dipolar condensates, and has important implications for ongoing studies of equilibrium and non-equilibrium dynamics in these systems.