How do I put out burning teammates?
Have the burning member run through a tile that contains water. This will put out your soldier and remove the Burning debuff.
This can include tiles such as Fountains on certain tilesets or puddles, even the green goo encountered in the first Blacksite mission you go through. If the weather is raining it will not put out the fire.
Medkits will also remove any status debuff, like Burning or Poisoned.