It all depends on the vehicle in question as to whether it saves fuel, and/or is detrimental to the auto tranny.
As long as the engine is running, it is rotating the torque converter, which in turn is pumping fluid around the auto box. This keeps all the bands, clutches, servos etc lubricated no worries.
When going back into gear, if you're not rolling up to a stop sign or red light or similar, it may help to increase the engine revs a little to match the revs the vehicle would be doing at that road speed. This will reduce any shock loading on the internals.
As for saving fuel, it depends on the vehicle, carbie cars, and early injected vehicles, it will definitely save fuel.
On later vehicles, with more sophisticated electronics, Your mileage may vary. Some vehicles will cut fuel to all/most injectors in a closed throttle position. Where as at idle, and in neutral, all cylinders are getting fuel to keep the engine running smoothly.