It is prohibited by Torah law to turn on an electric light bulb or activate an electric heating element, because when the bulb is turned on or a heating element activated, the filament incandesces (see above, 4:5 and n. 3, regarding fluorescent bulbs), and this is a form of Hav’ara prohibited by the Torah. While it is true that in the past, people did not generally produce heat or light by heating metal, we nevertheless find that heating metal for any purpose was prohibited by Torah law. Thus Rambam writes that if one heats metal with the intention of tempering it afterward by plunging it into cold water, he transgresses the Torah prohibition of Mav’ir (MT 12:1). Clearly, then, heating metal is forbidden by Torah law if it accomplishes some purpose.
Extinguishing a light bulb or heating element is only rabbinically prohibited. It is different from a normal act of Mekhabeh, whose purpose is to create charcoal (as explained above 16:5). In contrast, turning off an electrical appliance creates nothing, so its prohibition is rabbinic.
The Torah forbids performing prohibited melakhot by means of electrical appliances. Thus, activating an electric flour mill on Shabbat constitutes a violation of Toĥen, operating an electric mixer on Shabbat violates Lash, and so forth for all melakhot. Although the person is not manually grinding or kneading, since he presses the button or flips the switch to turn the appliance on, all melakhot performed by the appliance are viewed as melakhot performed by him (Oraĥ Mishpat §70; Aĥiezer 3:60).
Just as one may not turn on an electrical appliance by pressing a button, so too one may not activate something by remote control. Even though there is no direct contact between the finger pressing a button on the remote control and the appliance, since this is the normal way to turn on such appliances, it is forbidden just as manual activation is forbidden.