After destroying a block that is on fire, the fire is still there. However, it turns invisible. To recreate this, set the top of a block on fire, then break the block. Now, walk over the area where the fire was. You will catch on fire, and you can hear the fire crackling. To put out the fire, you must replace the original block (which, in turn, replaces the fire visually) and proceed to put the fire out as normal. An alternative is to replace the spot where the fire was with a block. This will also get rid of the fire.