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To have a fire, you need 1) oxygen, 2) fuel and 3) heat. You can have different fuels that burn at different temperatures, but you always need some amount of heat to have a fire.
If you can use any fuel, what is the lowest temperature at which you can have a fire with an open flame? Is it possible to have a fire that's so cold that you can hold your hand in it without burning yourself?
Fires which burn below about 400°C are known as cool flames, and this is a well known and widely researched phenomenon. Most hydrocarbons and many alcohols can produce cool flames, and the conditions under which they burn depends on the oxygen content available. The most common encounter that most people have with the cool flame effect is through incomplete combustion of a petrol engine, leading to engine knocking.
The lowest recorded cool flame temperatures are between 200 and 300°C; the Wikipedia page references n-butyl acetate as 225°C. You can read a lot more about cool flames on that page. One thing to note is that cool flames are very hard to visibly detect at lower temperatures - both heat and light being two of the by products of the combustion process.
NASA have conducted a number of experiments in space on low temperature combustion of nano droplets of heptane, and this is an interesting area of science still being investigated.
An interesting point to make relates to your question of whether you could hold your hand in the flame and not burn yourself. Clearly the answer is no, but for some cool flames it may take noticeably longer to feel the pain of heat. Paper starts to discolour at about 150°C, but may not burn as ignition temperature is between 220-250°C. For reference, candle flames are about 600-1400°C.
Neither of the previous 2 answers mention a phosphorus flame, which behaves visibly like other flames we are familiar with, but burns at a temperature so cool that you can hold your finger in the flame without getting burned. This only works with white (not red or black) phosphorus. It is (with proper precautions!!) placed into the chamber of an alcohol burner or similar device with the wick removed from the vent tube and warmed gently on a hot plate. The phosphorus (P4) vapor will, if ignited (maybe spontaneously), burn with a very cool flame. The same holds true for phosphine (PH3)and related low molecular weight phosphorus hydrides.
Very late to the party, but $\ce{PSF_3}$ (thiophosphoryl fluoride) burns with a cool flame. As reported in wikipedia (https://en.wikipedia.org/wiki/Thiophosphoryl_fluoride), $\ce{PSF_3}$ was discovered in 1888 by J.W. Rodger and T.E. Thorpe. As stated on the wiki page: "The discoverers were able to have flames around their hands without discomfort, and called it 'probably one of the coldest flames known'." It spontaneously ignites in air, burning with a "greyish-green" flame and producing "solid white fumes". The wiki page also notes, rather dryly, that $\ce{PSF_3}$ "is useless for chemical warfare as it burns immediately and is not toxic enough." Always some pesky drawback ruining what could have been a very sweet lecture demo.
It depends on your definition of fire. You can have hand warmers based on iron filings, the iron oxidises slowly and gives off heat as it does. So you have a redox reaction and heat is given off, but its not enough to burn a human hand.
http://www.hometrainingtools.com/a/homemade-hand-warmers
If a "fire" has to involve actual flames or smoke, then of course that's different.
I don't really think there is a measurable temperature where you can stack a block of $\ce{ClF3}(s)$ on top of a block of $\ce{LiAlH4}(s)$ and not worry about yourself and the room disintegrating due to an immediate and violent conflagration. Certainly nothing happens at ${0 K}$, by definition in fact, but at what miniscule temperature you have to find your running shoes, is hard to tell.
