When a spy is “burned”, it means that he or she has been discovered. For example, when you are tailing someone, and this person notices that you are tracking their moves, then you’ve been burned. Even worse, when you’re exposed as a spy, you’re also burned.
The word “burn” is also used in combination with other words, like “burn bag”. This is a bag that holds important information that should be destroyed immediately on discovery. That these items are kept together makes it easy to dispose of them quickly.