The frequently requested feature is reference in today's Hearthstone patch.

Since 2012, Blizzard fans have been requesting a way to appear offline when using the developers desktop application. While Blizzard has promised this on World of Warcraft, Starcraft II, and Diablo III, there's been no word of its arrival on other services. Separately, Overwatch's Jeff Kaplan has expressed his desire to bring it to his game's service. Today's Hearthstone patch contains references to messages that pop up when you attempt to interact with someone despite appearing offline, indicating that it may be arriving soon.

 

  • GLOBAL_FRIENDLIST_CHALLENGE_BUTTON_IM_APPEARING_OFFLINE You can't challenge {0} right now. You are set to appear offline.

  • GLOBAL_CHAT_SENDER_APPEAR_OFFLINE You may not send chat messages while you are set to appear offline.

 

The original announcement from 2012 can be found below followed by Jeff Kaplan's post from November of last year.

Originally posted by Blizzard.

In the coming months, we're planning an update to Battle.net that will give you more control over your online presence when playing Blizzard games. Soon, World of Warcraft, StarCraft II, and Diablo III players will be able to select "Appear Offline" as one of their Battle.net social-status options (along with Available, Busy, and Away) for those times when they want to wander Azeroth by their lonesome, dominate the galaxy in radio silence, or slay demons in solitude. When you choose this option as your status, all of your Real ID friends, BattleTag friends, and character-level friends will see you as “Offline” in their friends list whenever you’re logged into a game.

The “Appear Offline” status option will be added to World of Warcraft, StarCraft II, and Diablo III separately in future updates for each game. For Diablo III, we expect to add the feature in a patch after 1.0.5. For World of Warcraft and StarCraft II, we expect to add the option sometime after the release of each game’s upcoming expansion.

Thanks for your continued feedback, and we’ll share more information on the availability of this new feature in the months ahead.

 

Originally posted by Jeff Kaplan.

We would like to add an "Appear Offline" option to Overwatch. We are in the process of coordinating this with Battle.net and the other game teams. Because our social system is a cross-game system (WoW, Hearthstone, Heroes, SC2, D3) adding this functionality is not trivial for us. I know there has been discussion around this feature for a long time, and things have definitely not moved at a rate which is acceptable, I can assure you we care about the feature and want to figure out how to make it happen.

For your particular issue regarding invites, you can go into the Social Options menu and turn off invites.

Thanks for being patient with us.