Is it allowed to make distilled water on the Sabbath, i.e. to begin before the Sabbath and let the machine continue to work even after nightfall?
The distiller works by boiling water and then condensing the steam. This makes very pure water.
A process close to distilling water, is condensation from an air conditioner (doesn’t involve boiling, but it creates water from evaporated molecules). The water from an air conditioner is muktzah.
From the OU website;
The dripping water is muktzeh because it is classified as nolad gamur, newly created on Shabbos; Rav S.Z. Auerbach (Shemiras Shabbos K’hilchasah 13, note 130); Ashrei ha-Ish 17:221; Shevet ha-Levi 9:128; 11:204.
Based on the comments, that perhaps the water that is made after Shabbos started is batul in the water that was made before Shabbos, the Shevet Haleve mentioned above 9:128 says that water would only possibly become batul in wine or some different liquid, but not water in water, as it is min b’mino.