The Diablo III Website announces that a pretty basic feature is soon being added to Battle.net; the ability to appear offline. This will actually be patched into each game individually, starting with the upcoming version 1.0.5 patch for Diablo III:

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.