If the flame goes out on a stove on yom tov, may one turn the dial to shut off the gas? If so, with a shinui or can it be done in a normal way?
The tap can be turned off in the normal way.
If the tap was not fully open, the tap will not be muktzeh, because the flame can be turned up on Yom Tov.
Even if the tap was fully open, in which case there is apparently a problem of muktzeh because of the prohibition of turning the flame down, the Shemiras Shabbos Kehilchasa (13:13 note 61) writes that on account of the possibility of having the flame turned out in a permitted manner, and possibly also on account of the danger involved, the prohibition of muktzeh will not apply.
Elsewhere, the Shemiras Shabbos notes that it might also be permitted to turn off the tap (in a normal way) on account of the halachah of geraf shel re’i (Chap. 1, note 76).