While the other answers are accurate, I actually have done this.
I shut down fallout shelter and did not reopen it for 4 weeks. I left my explorers in the wilderness, and some survived!
They were not anything fancy, just some level 1s with terrible stats that I sent out to die. But they somehow survived. I am bringing them back to the vault and renaming them Bear Grylls and Les Stroud.
I also had some dead bodies lying around that did not decay after 24 hours.
My theory for all of this is that the game does not actually run in the background, it just saves timestamps, then calculates what happened the next time you open up the app. That means that even though the explorers were dead when I opened the game back up, they had only died a few moments ago rather than several weeks ago, so the 24 hour body decay hadn't completed yet. It's kind of a shrodinger's cat kind of thing, where the dwellers are both dead and not dead until you start the app back up. But I digress.
I noticed that the survivors didn't have any negative events in their logs when I started back up. This might be a bug or something that the devs never tested because waiting a week to run a test case is a bit of a challenge.
In any case, I would not recommend that you have any explorers out when you shut down for two weeks. I don't know if my experience was a fluke, or happened because I upgraded the app before I started up again, or what. The others have mentioned this in more depth, but here is the quick list:
- Recall your explorers
- Or leave them out there FOR SCIENCE (and let us know the results)
- Top off resources
- Make sure you have some wiggle room between your total resource capacity and the vault's required resources (vertical line on the resource bar) so that all resources are satisfied after the app takes its little bite when it shuts down. Otherwise your vault will be a mess with damage/radiation/lower power when you get back
- Make sure any dead bodies are cleaned up
- You don't want everyone to be unhappy on your return
- Put dwellers in training
- They might as well use the downtime to gain a stat
Your vault will not suffer from fires, raider/deathclaws/molerat/rad roach attacks, or other events while the app is shut down. Events should still queue up for explorers, unless your guys hit the same crazy survival bug that mine did. The safer thing is to recall them.