Appearing offline will show you as offline to everyone in your Blizzard friends list both in the Battle.net app and in-game. Gameplay and game-specific features such as matchmaking are unaffected while you are appearing offline. Note that in the case of World of Warcraft, you will still appear online as your character in-game.

While appearing offline you can still send and receive messages in the app, but it is up to you when you want to respond.