With shotguns dry firing is going to be much less beneficial than with a pistol because of the trigger weight vs. gun weight ratio. The reason, or one of the reasons, you dry fire a pistol is to focus on the front sight and see if your trigger finger/grip is effecting it in some way (ie: your sights dip low-left when you pull the trigger) because a 5 pound trigger pull in a 2 to 3 pound pistol is huge while on a shotgun you have an 8 to 10 pound gun with a 6-7 pound pull which is not nearly as much comparatively. Not to mention %95 of the time a shotgun is not intended as a precision weapon.
Mostly what I do with my shotgun is work on mounting it quickly from different positions, including off a table and working the safety, weak hand reload with par timer, work on transitioning from target to target without worrying about pulling the trigger, and transitioning either to the shotgun from another weapon or vice versa.
I think outside of loading drills the biggest time saver you can work on outside of the range is going to be mounting the shotgun consistently under pressure. Sort of like working your draw, if you can acquire your sights and break that first shot off quickly you're going to be doing better than half the crowd right out of the gate. If you look at Keith Garcia's "Load 12 Challenge" I think he mounts the shotgun and hits a plate at 15 yards in .4 seconds. Compare that to the average competitor and he probably saves a second on the first shot.
CM