G-d forbid, but let's say that my house catches fire either on Shabbos or a Yom Tov. Can I call 911 or must I find a non-Jewish neighbor to call? We have an eruv. Can I take my car keys, cell phone and wallet with me? If the fire is still small, can I do anything myself to put it out?
The laws of a home fire on Shabbat are discussed in Shulchan Aruch OC 334.
In a case where there is a fire which is not even possibly a danger to human life, one may not extinguish it on Shabbat. Moreover, by rabbinic decree one may not remove his possessions from the house, lest he become overwhelmed, forget that it is Shabbat, and come to extinguish the fire (:1). What he is allowed to save is:
- Food for the remaining of the 3 Shabbat meals for his family and animals (:1).
- If he the food is all in one pot he can remove more than the minimum if it is in the same pot (:6).
- He can remove along with this any vessels he needs to use for the remainder of Shabbat (eg. cups) (:8).
- On Yom Kippur one can save one meal's worth of food for dinner after the fast (:4).
- A much clothing as he can wear at one time. He can then take all the clothing off outside, go inside and repeat (:8).
- Books with Torah content (:12).
- Some exclude the book of Esther (when not written properly on parchment), as it has no mention of God's name (:13).
People living in adjacent houses who fear the fire will reach them in the future may save whatever they want (:1). Additionally, he is allowed to call all his friends to come save whatever they want, but if they do so, they acquire the objects and don't have to return it to the original owner after Shabbat (but may return it if they wish) (:9).
One can call over a non-Jew without telling them to put out the fire even if one knows that the non-Jew will anyway. One cannot allow a Jewish minor to put out the fire. (:25)
The Shulchan Aruch limits the above discussion to a place where there is an eruv, but the Rama extends it to any location in which the prohibition of carrying is only rabbinic (:10).
What one can do to avoid financial loss is place container full of liquid or wet clothing in a place the fire has not yet reached even knowing that the container will break and put out the fire (:23-24).
In a case where there is a fire which is even possibly a danger to human life (the vast majority of modern cases), then it is a big Mitzva to violate any Shabbat law in order to save lives including extinguishing the fire directly and calling a firetruck. One should even perform this Mitzva with alacrity (Rama :26). If all else is equal, one should have the Jew with the greatest standing do this Mitzva to publicize its importance (OC 328:12). I have not found discussion regarding if the above mentioned prohibitions apply even when the fire is dangerous enough to allow you to put out the fire yourself (obviously, assuming they do not directly prevent saving lives).
A home on fire is a danger to life. You can, halachically, do very nearly anything necessary to preserve human life, including violating all the Shabas or yom tov prohibitions, and thus including calling an emergency number or doing what you can to extinguish the fire. The second it takes to grab your keys or wallet is a second of risk, which would mean you shouldn't do so.
This answer assumes that a home fire is a danger to life, which in my opinion is an assumption one must make in the very, very vast majority of cases. For more of the theory behind home fires in halacha and what to do in the very rare case that the home fire is not a danger to life, see Double AA's good answer.
The proximity of gas lines to all modern homes makes every major fire a hazard for the entire community. One who is stringent on safek Pikuach nefesh should be blessed!
If there are many seforim in the house - too many to remove safely - one may ask a non-Jew to call the fire department/put out a fire. Similarly, one may ask a non-Jew to call the fire department in order to save Mezuzot. Yet another reason to have many seforim and Mezuzot on all your doors...
Source: Shulchan Aruch 334:18 and M"B ad loc.