It would be a lot of active characters for the server to track. It would amount to every character owned by every player in the whole game being online at all times. They wouldn't eat up bandwidth, but the server would have to run all their automated tasks, eating server resources at all times of the day and night.
From a player perspective, it would be weird to see a friend's character in "zombie mode", offering robot responses to stimulus and not remember the great dick joke from last time you quested together. You'd never see your own guy in the game, unless you were playing on another character, which might be vaguely gratifying, but not terribly productive.
Also, if the benefits of offline training or commerce were substantial, it would encourage players to build a stable of automatons, setting their drones to work leveling up or farming resources or upgrading in-game assets, rather than actually logging in and playing the game. I know that when I played EvE online, there would be months when I'd log in a few minutes a week to update market orders and skill training queues, never so much as undocking my battleships. Not the most engaging gameplay model.
How much of your game's infrastructure would be tied up with this? If the best alchemist in town sets his character to level up archery for a week while he's on vacation, will the potion market be seriously impacted? Will an NPC take his place? Will it be easy for another character to fill the gap? Will my automatic work when I'm offline be an oppressive duty, or a meaningless bit of roleplay?
I think this kind of idea works better int he context of a single-player game. What if I could get my guy in Skyrim leveled up in smithing, then buy a shop in Markarth, then retire to make sweet axes and cheap arrows? My next character could go there and purchase high-end gear from him, maybe get some training in the skills the older character had mastered, or barter for rare items that I had collected in my earlier travels.
What do you hope to achieve with this?
Ambiance? Because characters in online games are created ex nihilo, usually through a customization interface, it's likely that you'd wind up with cities packed with idle characters, at least a few of whom would have immersion-shreddingly horrible features and dress. As I said above, the inability of the drones to recognize friends or interact with active players in a meaningful way would seriously reduce the value of their presence.
Is it productivity you're after? Anything a robotic Character player model could do could be done--more cheaply--by simple offline timers. Tell him to build 40 leather backpacks. You get a progress bar and nobody in-game needs to see your little man pathing from the tannery to the leatherworks to the tailor for hours on ends, hanging up in doorways and staring straight ahead and ignoring my chat hails.
Hello Iron Chef Carnage, the reason I am proposing to push the MMO boundary is that I feel it would enhance games. Not all characters would be out, some could be housed, or have the option to log off, but being online even when the player is offline allows for something else I was avoiding explaining. Then the character could be controlled somewhat through a smart phone app. And Texting the character would show up like texting a friend. So if your character needs help, you log on through your smart phone and alter what is needed.
As for a stable of automatons, give the player a limit to how many characters they can have online at once, and if they do nothing but level up,then so be it, but I don't plan to have characters that are WoW clones where they can master multiple crafts and not need anything or anyone else. If you make a resources region specific like real, make resource gathering, refining, and crafting its own game that requires interaction with many other players then if you do have the best baker, they are gonna need flour from someone who milled it, grown by a farmer, eggs from someone, pottery for bowls, and so on. Thinking that every game is like WoW will mean that we never advance the game society or the developers beyond expecting stale bland games.