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.Or leave them out there FOR SCIENCE (and let us know the results).The others have mentioned this in more depth, but here is the quick list:

I don't know if my experience was a fluke, or happened because I upgraded the app before I started up again, or what. In any case, I would not recommend that you have any explorers out when you shut down for two weeks. 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. I noticed that the survivors didn't have any negative events in their logs when I started back up. 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. 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. 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. I also had some dead bodies lying around that did not decay after 24 hours. I am bringing them back to the vault and renaming them Bear Grylls and Les Stroud. They were not anything fancy, just some level 1s with terrible stats that I sent out to die. I left my explorers in the wilderness, and some survived! I shut down fallout shelter and did not reopen it for 4 weeks. While the other answers are accurate, I actually have done this.

Fallout shelter broadcast center levels