The King, King's Guards and Banes do not hold cards and cannot take advantage of this dynamic.
Players only have the option to burn cards within the bounds of the burnout timer, however, once you roll your dice, the timer will end for you and you will no longer be able to burn any more cards. In combat, the sun or moon symbol in the center of the timer correspond with the current day cycle and tells you whether Suns or Moons will be hits, and the other misses.
In combat, burning cards can offer guaranteed hits, shields, explodes, or simply a mechanism to get rid of unwanted cards by the end of one's turn.
When facing Perils, players can match the symbols on the screen with the symbols on the card for a better statistical chance at (or guarantee of) clearing the Peril. This is especially important when attempting to survive Palace Perils, where the odds of clearing by Dice rolls alone are incredibly slim.