I searched a little while for this issue and I didn't see this same problem. But sorry if someone wrote it already. I am on a touch screen windows computer 

I was playing on a superflat world with Fire Spreads setting off.

To test the setting out, I set a few pink wool (floating in the air) on fire with flint and steel.

After waiting for some time and seeing that the fire didn't spread, I broke the blocks.

The fire sound effect continued to play long after the wool was broken and it doesn't seem to be stopping. I exited the world (save&quit) and rejoined and the sound still played. The same happened when i restarted the game. I also marked the area of the sound with a fence, went really far, then returned to the spot and it was still there. Also turned the Fire Spreads setting on then off again

When you go away from the fire sound, it fades out from the distance

Afterward, I turned Fire Spread on again then set a jungle wood plank (on the ground) on fire with a flint and steel. After it burned away and all the fire from that disappeared, the sound stopped. And again, the sound from the Fire Spread off fire was still playing. 

I also tried throwing splash water bottles and dumping water buckets all over where the first fire was but that didn't work either. At first the splash bottles caught on fire indicating that they touched where the fire once was but the sound still didn't go away

Then I turned the setting off again and started another fire but this time stamped out the fire itself and not the block under it and the sound went away. I tried punching the air in the area of the first fire but that didn't stop the sound either.