What happens with ammonia ions in tap water after they are separated from chlorine?
Chloramines (mono-, di-, and tri-) are used as a disinfectant for drinking water treatment (the source water, not the tap water) - this process is called combined chlorine. Upon contact with pathogens, organics, etc., chloramine is reduced to chloride (not chlorine), ammonia, and hydroxide. However, this half reaction is for alkaline conditions. The pKa of ammonia is around 9.4, so around circumneutral pH, some ammonia will exist as ammonium, the protonated state.

So, will activated carbon (AC) remove the products? First, we need to clarify something about AC - it is not catalytic, it is a physical adsorption removal process. AC will remove chloride ions, but it will not remove ammonium. Ammonia is quite volatile and will move from the water to the gas phase if e.g., a pitcher of water is left at room temperature. If you wanted to remove ammonium as well you could use something called biologically activated carbon (BAC). One last note, chloride will not "bind" will AC, it will be adsorbed through electrostatic forces.