Apple - How to quit without saving using just the keyboard?

The standard shortcut for "Don't Save" in the dialog you mention is command ⌘+Delete. The old shortcut, command ⌘+D, can be restored by issuing the following command in Terminal:

defaults write NSGlobalDomain NSSavePanelStandardDesktopShortcutOnly -bool YES

and to revert back,

defaults write NSGlobalDomain NSSavePanelStandardDesktopShortcutOnly -bool NO

In Microsoft Office 2011 for Mac, command ⌘+D seems to work by default (but the standard shortcut does not).

Basically, try either command ⌘+Delete for Apple or standard applications, and command ⌘+D if that does not work.

It's not the default behavior for tab–I read because someone at Apple felt tabbing "wasn't magical enough"–but in the Keyboard preferences, click the Shortcuts tab. Then select All controls to make the Tab key behave the way you want. Or just press control+F7 to toggle this.

If you are at the following dialog, just hit d...no key-combo necessary.

Since Lion, the standard key command for Don't Save is Cmd ⌘ Backspace ⌫
Prior to that it was Cmd ⌘ D which some apps still respond to.

Ctrl Tab will switch between the options, which can then be actioned by hitting Spacebar , only if you have All Controls set in System Prefs > Keyboard > Shortcuts