After you close the game, your properties are still generating rent for 4 hours. Assistant extends this to 12 hours. Once purchased, it will stay on your account forever.
If you lost your rent, you can use "recovery" option for rent from last 12 hours. It is available after you purchase assistant and in exchange for coins you can recover rent from hours you were inactive (extending 12 hours of collection).
For example, if you were inactive for 16 hours, you will get 12 hours rent with assistant and can use "recovery" to get rent from 4 hours (for coins).