There is a wide consensus that the ubiquitous presence of magnetic reconnection events and the associated impulsive heating (nanoflares) are strong candidates for solving the solar coronal heating problem. Whether nanoflares accelerate particles to high energies like full-sized flares is unknown. We investigate this question by studying the type III radio bursts that the nanoflares may produce on closed loops. The characteristic frequency drifts that type III bursts exhibit can be detected using a novel application of the time-lag technique developed by Viall & Klimchuk (2012) even when there are multiple overlapping events. We present a simple numerical model that simulates the expected radio emission from nanoflares in an active region, which we use to test and calibrate the technique. We find that in the case of closed loops the frequency spectrum of type III bursts is expected to be extremely steep such that significant emission is produced at a given frequency only for a rather narrow range of loop lengths. We also find that the signature of bursts in the time-lag signal diminishes as: (1) the variety of participating loops within that range increases; (2) the occurrence rate of bursts increases; (3) the duration of bursts increases; and (4) the brightness of bursts decreases relative to noise. In addition, our model suggests a possible origin of type I bursts as a natural consequence of type III emission in a closed-loop geometry.