1. Life-threatening situations

If there is even a doubt that a life may be at stake (e.g. the fire may spread to other buildings), it is permitted to put out a fire in any manner, call the fire department, and break any other necessary Melachot. Even if it becomes clear that one's help was not needed, his efforts will nonetheless be rewarded.

2. Physical danger to people

If it is certain that no lives are at stake, but people might get hurt physically (e.g. children or elderly people could panic and fall down stairs), it is only permitted to extinguish the fire outright through the breaking of a Rabbinic law.

If possible, the best solution is to carry the fire to a place where it cannot spread (e.g. to a bathtub), since this only violates a prohibition of Muktzah. It is preferable not to take the fire outside, because the extra air supply can increase the fire. If necessary, one could even pour water to extinguish the fire, as this is a Melacha Sh'aino Tzricha l'Gufo

In all cases where lives are not at stake, it is best to ask a non-Jew to do the Melacha.

3. No physical danger

If it is certain that nobody will be hurt by leaving the fire alone, it is prohibited for a Jew to extinguish the fire in order to save material belongings, based on the idea of serving God B'chol M'odecha -- "with all one's belongings."

Exceptions (i.e. when certain actions may be taken against the fire):

a. In the case of a minor, but definite, monetary loss, it is permitted to:

(1) Hint to a non-Jew to extinguish the fire, provided that no prohibition is violated in the process of telling the non-Jew.

(2) When no non-Jews are available, a Jew himself may put out the fire, but only indirectly (Gramma). For example, one may pour water around the fire, so when the fire spreads, it will go out by itself. (When dealing with a tablecloth or a carpet, one should use colored liquid like dirty water or red wine, because of the Melacha of M'labein.)

b. In a case of great loss, or for the prevention of a definite, substantial monetary loss, it is permitted to:

(1) Directly ask a non-Jew to put out the fire.

(2) Move Muktzah -- e.g. the fire itself, or one's possessions if articles would be damaged by leaving them where they are. For example, if a candle falls out of its holder onto the table (which can result in great damage), it is permitted to pick up the candle and slowly return it to the holder. Also, one may move a Muktzah object away from the fire to prevent it from getting destroyed.

Gezeira: Worry for monetary loss

In a case where the fire must be left alone, the Rabbis were afraid that a Jew, worried about his belongings, might safeguard his belongings by taking them out of his house, and might even forget that it is Shabbat and put out the fire or do other Melachot.

Therefore, if one particular apartment in an apartment building is on fire, the following Halachot apply to every other apartment owner in that same building: (For the burning apartment itself, the exceptions listed above apply.)

a. It is prohibited for the resident to remove possessions out of his private apartment (even if he is able to do so without breaking the prohibition of transferring). This becomes prohibited because the Rabbis invalidated the workings of an Eiruv for such a person.

b. It is permitted, however, for him to:

(1) Remove [to a location within the Eiruv]:

(a) Food for the remaining Shabbat meals (and their necessary vessels),

(b) Torah books (which one may even ask a non-Jew to carry outside an Eiruv), and

(c) Clothing, in the way it is normally worn (even by wearing many garments on top of each other, and even to a place where there is no Eiruv).

(2) Tell other Jews (who don't live in the same building) to take the person's belongings out of the burning building (to a location within an Eiruv) and to keep them for themselves. Since they are not personally involved, they are not likely to put out the fire. It is a righteous act for these Jews to return the items to their owners after Shabbat.

(3) Hint to a non-Jew to remove property to any place, by saying, "Whoever saves my property will gain" (or, "will not lose"), and then reward the non-Jew after Shabbat.

(4) Within a private domain/house, one is allowed to move property (including Muktzah) away from the possible effects of the fire.

It is advisable to survey one's house and plan what to do in case of a fire on Shabbat (God forbid).