Selecting the "Don't Save" button using the keyboard
The default keyboard command for Don't Save is Cmd ⌘ Backspace ⌫ which was introduced with Lion
Prior to that it was Cmd ⌘ D which some apps will still respect, but not all.
very late addition 2021 - Adobe apps still don't like Cmd/⌫ & only respond to Cmd/D
Ctrl Tab will switch between the options, which can then be actioned by hitting Spacebar , but only if you have All Controls set in System Prefs > Keyboard > Shortcuts
Imported from my own answer at https://apple.stackexchange.com/questions/154046/how-to-quit-without-saving-using-just-the-keyboard