Mining a block on fire does not put the fore out. If you walk to where the fire was you take fire damage. I discovered while bed mining in the nether. I’ve replicated the issue in the over world and the nether with stone, netherack and wood.

to replicate:

1) place block.

2) light the block on fire

3) mine the block, the fire will not be visible.

4) walk to where the block was, you will take fire damage.

5) replace the block and it will be on fire.