The Chazon Ish (Kuntres ha'Muktzeh 41:16) writes that the flame is Muktzeh because normally one would not move it from its place. Rav Shlomo Zalman Auerbach zt"l says that based on this one may carry an electrical appliance which has a heat element since it is made to be transferred from place to place.

Rav Shlomo Zalman himself (Minchas Shlomo 14) suggests that fire is intangible and that is why it is Muktzeh, since at any given moment it is constantly changing and morphing into something else. Therefore, a heat element which remains constant would not be Muktzeh.

Rav Moshe Feinstein zt"l (Igros Moshe, Orach Chayim 3:50) suggests that a flame is not a utensil designed to serve a person and therefore it is Muktzeh Machmas Gufo.

Some views understand this as a self-evident Muktzeh, since people tend to keep a distance from flames and not to touch them, so they are not viewed as something usable (Rav Shmuel Auerbach zt"l in Orchos Shabbos 19:241). Based on this, the same book prohibits moving a heater on Shabbos since one would normally not touch it due to its heat.

Yoel Domb