With all the potential saving of goods one may do if a fire breaks out on Shabbat, the one thing you cannot do is put the fire out. However, the Mishnah states that if a non-Jew comes to put out the fire, a Jew is not to say “extinguish it” or “do not extinguish it” – because he is not obligated to rest on Shabbat and we are not obligated to make a non-Jew rest.
However, the rabbis allow an implication:
R. Ammi said: In the case of a conflagration they [the Rabbis] permitted one to announce, 'Whoever extinguishes [it] will not lose [thereby].'
Suggesting, but not stating, that a reward will be forthcoming after the Shabbat (of course, one DOES have to fulfill the implied reward!)
A story illustrates:
Our Rabbis taught: It once happened that a fire broke out in the courtyard of Joseph b. Simai in Shihin, and the men of the garrison at Sepphoris came to extinguish it, because he was a steward of the king. But he did not permit them, in honour of the Sabbath, and a miracle happened on his behalf, rain descended and extinguished [it].
In the evening he sent two sela' to each of them, and fifty to their captain.