Some general examples: Any active navigation keeping track of your location if you switch apps. Pebble monitoring notifications to be sent to a smartwatch. Email or chat client periodically checking for messages.

There is a Developer Setting in Android to instantly kill off-screen apps, but that would make multitasking a hassle.

On that note, what I have witnessed is the opposite, phone OS being overly eager to put apps in sleep / idle mode, to the point of ignoring user settings of “do not optimize battery” and “allow background processing”. Many (most “normal”) apps handle notifications through Google, but many FOSS and independent apps need to run in the background to check for messages, updates, etc.