The topic of extinguishing flames on Yom Tov is too extensive to cover thoroughly in this forum. We will discuss the basic sources and some practical options.
The gemara[1] says that according to the Rabbanan, one may not extinguish a burning block of wood on Yom Tov so that a house or a pot does not become smoky, nor otherwise directly extinguish a fire just to prevent a modest loss. However, it is not clear whether we follow the Rabbanan’s opinion and to what particular set of circumstances this prohibition applies. Some Rishonim[2] say that the gemara refers to a case in which the smoke will not severely hamper one’s Yom Tov activities (e.g., he can spend quality time in another house); if extinguishing the fire is necessary for Yom Tov needs, it is permitted. After all, one may perform a variety of melachot, including havara,[3] for Yom Tov purposes. Others contend that even if extinguishing a flame enhances Yom Tov, it is still forbidden because the direct benefit of that action is only the removal of a problem, and this is not deemed a positive benefit.[4]
One of the differences between the opinions pertains to a case in which a food requires cooking or reheating with a lower intensity flame because the current flame will cause it to burn. If a smaller flame is available, the pot should be moved there, and the larger flame should be left alone. However, if there is no smaller flame, the Rama[5] rules that one may lower the flame so that the food does not get burnt. However, there are several limitations on this ruling. The Mishna Berura[6] argues that the situation must be such that the food will be harmed seriously if the flame is not lowered; lowering the flame to mildly preserve the food’s taste is forbidden. The Magen Avraham[7] and Mishna Berura[8] seem to say that if a smaller flame could be lit from an existing fire, one may not lower the larger flame. While the Igrot Moshe[9] argues cogently that lowering a flame (at least a gas flame) is no worse than lighting a new one, many Acharonim disagree.[10] The Shulchan Aruch[11] maintains the position that lowering the flame is never an option, although significant Sephardic poskim consider leniency in cases of great need and/or mitigating halachic considerations.[12] (There is often a practical option of moving the food somewhat away from the flame.) In any case, other halachic options should certainly be sought.
Even on Shabbat, g’ram kibbuy (indirect extinguishing, e.g., placing a noncombustible material around a fire so that it goes out instead of spreading) is permitted to prevent a financial loss.[13] It is debated whether the same rule applies to Yom Tov or whether on Yom Tov it is permitted without any special need. Therefore, many poskim discuss the following suggestion. One can boil a full container of water (as long as he will actually make use of the water and is thus boiling for a positive reason) so that it spills over and douses the flame. At that point, he can turn off the gas. This is considered indirect, in part because one does not touch the fuel and because the result is separated time-wise from his action. Grounds for leniency are strengthened by the fact that, unlike in the classic case of kibbuy, there is no leftover coal-like residue.[14] However, on both technical and halachic grounds, this is not the ideal system.[15]
The best system is to set up a Shabbat-clock device that shuts off the flow of gas at an appointed time. Gadgets are now marketed specifically for this purpose. One may rely on the opinions that he can set the timer mechanism on Yom Tov[16] before the flame has been lit.[17]