Antonyms for the phrase "fans the flames" would be any word or phrase that suggests the opposite of promoting or exacerbating a situation. Some possible antonyms include words such as "calms," "soothes," "pacifies," "mitigates," "restrains," or "quells." These words all convey the idea of reducing or minimizing the intensity of a situation rather than adding fuel to the fire. For instance, instead of fanning the flames of a heated argument, one could try to calm the situation down by listening attentively, empathizing with the other person's perspective, and focusing on finding common ground.