Corals are not involved in the nitrogen cycle, as far as I am concerned. By that, I mean that even if they do have a role in breaking down ammonia, nitrate, nitrite; they play an insignificant role.
The main players of the nitrogen cycle are: aerobic bacteria break down Ammonia into Nitrite, and Nitrite into Nitrate. Anaerobic bacteria then break down the Nitrates in the system into Nitrogen gas, which leaves the system naturally. The aerobic bacteria are found on the surface and in the shallower depths of the sand and rock. The anaerobic bacteria are found in the deeper depths of the sand and rock, where oxygen is limited.
Corals do not harbor the correct bacteria for denitrification; they harbor a form of phytoplankton called zooanthelle which have a symbiotic relationshipo with the corals. The zooanthelle act as a food source for the corals and use photosynthesis to feed themselves. The zooanthelle may use up some of the nitrate and phosphate naturally occuring in the system, but the impact is insignificant.