Fallout Shelter closes after being open for a few seconds

the problem with xbox 360 is indeed overhead the only solution that works for me is strip the plastic casing the console needs way more air than its desinged do suck in as soon as you remove the plastic casing and the console has room to breath you should not have a freeze anymore a fair WARNING never touch the consoles metal when you stripped it 9 out of 10 you get a shock hope this helps

The game does continue to run for a little while after you leave. This is to encourage players to play for extended periods of time instead of just quickly checking on their vaults. You need to make sure that you have enough resources to survive the period in which the game runs yet you aren't online.

I would check that you have enough storage. You might not even have the capacity to store enough resources to survive closing the game.