If I'm not going to be using it at all, I'll shut the lid and leave it in sleep mode, although I think I do really leave it in sleep mode for too long periods really. I'll often leave New Leaf idling for short periods while I do something else, as I often get people talking to me. I do also simply forget it's turned on, so it'll sit there about an hour before I realise and shut the lid.
um yeah I always turn it off when I'm not using it. I often watch youtube videos while playing so sometimes I will put my 3ds down for a few minutes but then I'll either turn it off or continue playing.
If I'm not straight-up playing it, I'm looking up something related to what I'm playing, so it'll be open and running next to me. I actually rarely have it off, just so if I go somewhere, I can get play coins. I also sometimes just forget it's open, so it just stays on while I'm out or something, haha.
Sometimes, if I'm too busy for a couple hours and can't save my file on a game, I'll leave it turned on for a little bit while plugged into the wall....
fo ex. while scrolling through the forums, going to eat dinner, etc i personally don't, mainly b/c my activity log would be way off and battery life aha
Luckily stuff here doesn't appear to have to be about ACNL- I don't leave mine on for the same reasons, I may sit it down for like 2 or three minutes if, say, I'm waiting for an animal to come by my house or something. Or if it's in the last few minutes of a tourney and I want to see the results. :3