Having a gentile turn on the heater on Shabbos:
If it is a great need, such as on a very cold day or if one has small children and it is even mildly cold, then it is permitted to ask a gentile to turn on the heat whether on Shabbos or Yom Tov.
May one ask a gentile to turn on the fan or air conditioner on Shabbos?
 Admur 276/15; See Nitei Gavriel 33/10; Piskeiy Teshuvos 276/18
 The reason: As everyone is considered like a sick person in relation to the cold. [Admur ibid]
 See Nitei Gavriel 33/10; Piskeiy Teshuvos 276/18
 Chelkas Yaakov 1/145; 3/139; Minchas Yitzchak 3/23; Beir Moshe 6/93; Mishneh Halachos 5/42
 Igros Moshe Y.D. 3/47; Devar Yehoshua 2/7; Vayaan Yosef 1/137; Or Letziyon 2/25-5; Rav Ekyashiv in Koveitz Teshuvos 1/32; See Piskeiy Teshuvos ibid footnote 135; See also however Admur 313/17 that heat is not considered a Choleh like the cold, but simply like one who is Mitztaer.