If one set the thermostat of a radiator to a moderate temperature before Shabbat, but on Shabbat realizes that it is hotter than he had expected, he may lower the setting of the thermostat once it has switched the radiator off. Thus, he ensures that the radiator will remain off for a longer period of time, and the heating element will work for a shorter period of time. However, one may not lower the thermostat while the radiator is on. Doing so makes the radiator change its state sooner, from being on and heating to being off and not heating.
On Yom Tov as well, it is proper to lower the heat when the radiator has cycled off. True, grama is permitted for any need on Yom Tov, but in this case, there is a reasonable chance that if one lowers the thermostat, the heat will immediately switch off, which means that he has turned it off directly, not via grama.
If, during the course of Shabbat, one wants a radiator to stay on for longer, he must wait until the radiator has cycled on, and the temperature of the radiator has reached at least yad soledet bo (at least 160 degrees Fahrenheit or 71 degrees Celsius). Then he may turn up the thermostat so that the radiator will stay on for longer. However, he may not do so if the temperature is below yad soledet bo, as it would be a violation of Bishul. Once the radiator has cycled off, it is not permissible under any circumstances to turn up the thermostat, because doing so causes it to cycle on, and it may do so immediately, which means he has turned it on directly (see Peninei Halakha: Shabbat 10:3 and 17:7).
In contrast, on Yom Tov, when there is no prohibition on lighting or cooking, one may turn up the thermostat on a radiator even if its current temperature is under yad soledet bo. Nevertheless, this must be done when the radiator has cycled on and is heating. Otherwise, adjustment may take effect immediately, which means he will have turned it on directly.
The laws pertaining to Shabbat and Yom Tov are identical when it comes to air conditioners and refrigerators with manual thermostats. When the compressor has cycled on, one may turn down the temperature, which will keep the refrigerator or air conditioner on longer. When the compressor has cycled off, one may turn up the temperature, which will keep the machine off longer (Minḥat Shlomo §10; SSK 23:24).
Of course, all of this assumes that there is no electronic display recording the temperature. However, if the thermostat is adjusted by pressing buttons to change the temperature, and this is shown in an electronic display (as is the case with many air conditioners), then it is prohibited, both on account of Kotev, and because each press of a button makes direct use of electricity.