The Elizabethan stage practices of Shakespeare's time would not have provided any information to the audience about weather (nor about time of day, location, etc., for that matter). Information about the weather in Macbeth comes from dialogue.
When the witches say that fair is foul and foul is fair, they indicate a stormy day, or a day involving some kind of bad or odd weather, unnatural weather; particularly when they follow this line by referring to fog and filthy air. The refrain also meshes with the choice of day the three shall meet again on as offered by the First Witch in line two of the same scene: "In thunder, lightning, or in rain?" Notice no fair weather option is offered.
Macbeth, or course, echoes the refrain just before meeting the witches for the first time in Act 1.3, again highlighting the weather.
More important than the physical atmosphere (the weather), however, is the refrain's contribution to the ideas of reversal in the play. Gender roles are reversed (Macbeth and his wife). Grass and hay eating horses become cannibals. People that should be and appear to be loyal are not (Cawdor, Macbeth, Lady Macbeth). Reversals, opposites are so common that even an honest and loyal thane like Macduff is suspected of being what he's not (disloyal) by Malcolm.
Little is as it should be in the play: what's fair is foul and what's foul is fair.
The famous lines "Double, double toil and trouble; Fire burn, and cauldron bubble" have much power in Act IV, sc 1. When the act opens, there is thunder, and the setting is in a cavern with a boiling cauldron in the middle of the cavern. Before the witches even begin reciting their spells, the atmosphere is dark and foreboding.
The couplet is used three times for power. The more they recite/repeat the couplet, the strong the potion will be. The repetition also gives power to the atmosphere because the more they chant it, the louder they get and the cavern fills with the weather, the bubbling cauldron and the chanting. It's very much a threatening scene.