I'm not exactly sure what the button under my pet (in the pet window) is supposed to do. It says either "fight" or "rest", and the only real difference I see is my battle capacity change numbers (meaning the buffs are turned off). Why would we want to put our pet to rest? After all, it's there to fight with us, isn't it? Do we put it to rest to gain xp, like we do, or just to disable it for some crazy reason? (I suppose to disable it would be the most likely, but I don't see why if we want any chance of killing stuff in the higher levels.)
The pet-action phase takes time, so if you are in a competitive clearing event (like Orc Rebellion or Faction Wars,) and you don't need the de/buff to win within the cooldown period, then it is advantageous to rest your pet... as long as you remember to bring them back out afterwards.
More rare are intentionally lowering your BC or fighting power (for whatever reason) or because using the de/buffs actually backfires, particularly on the rage/damage flow in a match. Similar to all the reasons you might not use the maximum number of buddies allowed to you at any given time.