Do apps use data on iPhones?

If I'm using an app, does it still use data even if I'm not on the Internet? Specifically, it's the sleep app. So it stays on all night. Is this using all my data?

No, it shouldn't be using your data unless your allowing it to send push notifications or use your location.
If you are worried about your data usage, you can turn it off within the settings or set a limit so it doesn't let you go over your data usage limit.

Apps use up memory on the phone. Both Physical and Virtual. If you are talking cellular data it depends. Like if your 3g/4g is on and some1 sends you a message on kik or facebook it uses up some data. Its not megabytes of data. AT the most it is kilobytes and that is photo messages.