How to effect players that triggered a command in Minecraft?

First, add an objective to track if a player is burning:

/scoreboard objectives add isBurning dummy

On a clock, run the following commands in this order:

/scoreboard players set @a isBurning 1 
/scoreboard players set @a isBurning 0 {Fire:-20s}
/effect @a[score_isBurning_min=1] instant_health 20 20

The way this works is that, first, everybody's isBurning score is set to 1. When not burning, a player's Fire tag is at -20s, so the isBurning score for those with a Fire tag at -20s is then set to 0, leaving only those who are burning still with an isBurning score of 1. The instant_health potion effect can then be applied to the players that are burning.

As a general tip, the /testfor command is a lot less useful than people seem to think. When using /testfor, you are essentially just asking "Does this entity exist?" and getting a yes/no answer without being able to do anything with the entity matched your query. If you want to be doing something with the entity (healing, killing, teleporting, etc.), you shouldn't use /testfor.

This can be done in just one command in Minecraft 1.13:

/effect give @a[nbt=!{Fire:-20s}] instant_health 20 20