Question: May one adjust the flame on a gas stove on Yom Tov?
Answer: While one mustn’t ignite a new flame on Yom Tov (Shulchan Aruch OC 502:1), one is allowed to light a new flame from an existing one or turn a flame up. This only applies to a gas flame - not to an electric stove - and may only be done for cooking purposes, etc.
R’ Moshe Feinstein (Igros Moshe OC 1:93; 1:115; 4:103) writes that one may turn the flame down to prevent food from burning though shouldn’t do so just to save money or to cool the kitchen down, etc. R’ Avrohom Blumenkrantz (Chasdei Avrohom 31) explains the rationale behind this. While one can’t turn down a wick or remove oil from a lit lamp on Yom Tov, turning down a gas flame is akin to not adding more fuel to a fire.
R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 13:10) writes that one shouldn’t turn the flame down unless there is no other flame, though R’ Moshe allowed one to do so even if one could light another smaller flame. The Shearim Metzuyanim Behalacha (98:13) writes that as it is dangerous to leave an extra flame on, there is no need to do so.