How to stop fire from going out without netherrack

Update to 1.13: the syntax and functioning of commands has changed (twice). It is now possible to combine all into one Repeat Always active command block:

execute unless block X Y Z fire run setblock X Y Z fire

where X, Y and Z are the X, Y and Z coordinates of the fire block, respectively

Breakup:

execute unless                  # unless (if ... is not)              
    block                       # the block at 
    X Y Z                       # the coordinates [X,Y,Z]            
    fire                        # is fire
    run setblock                # place a block
        X Y Z                   # at the coordinates [X,Y,Z]  
        fire                    # of fire

Resets the fire within 1/20th of a second after burnout, and is lag-friendly*. The command block can be anywhere. Here is a screenshot:

_{* as lag friendly as 1.13 can be}