What is the youngest age that a child can be that their tuning on the oven or burner would be considered bishul yisrael?
According to a number of poskim, any age will prevent bishul akum. However according to some as long as the child is old enough to put the wood in because he was told to by his father, that is also old enough. See sources.
See Rema Y:D 113-4,7 that fif the Yisroel placed wood into the fire, even if it was unintentional it still prevents bishul akum. Also see Darcei Teshuva 113-55 that even according to those who say that the Jew has to have intend to put it there to prevent bishul akum, this is only l’chatchila, however b’dieved even if the person didn’t have kavana it will prevent bishul akum. Also see Bais Dovid Y:D 113-4 that even a very young child can make it bishul yisrael as long as the child did it because the adult told him to. Additionally, see M’tziyon Teitzey Torah 267 that R. A Nebenzhal, R’ E. Auerbach and R’ Y. Chesner shlit”a hold that even a “shoita” that puts some wood on the fire will prevent bishul akum.