This paper investigates the development of fault detection and isolation (FDI) filters for distributed time-delay systems. The notion of a finite unobservability subspace is introduced for distributed time-delay systems and an algorithm for its construction is presented. A bank of residual generators is designed based on our developed geometric framework so that each residual is affected by one fault and is decoupled from the others while the H-infinity norm of the transfer function between the disturbance and the residual signals are kept at less than a prespecified value. Simulation results for a combustion system in a rocket motor chamber demonstrate the effectiveness and capabilities of our proposed FDI algorithm.