Is fire plasma? Is Fire a Plasma?
If not, what is it then?
If yes why, don't we teach kids this basic example?

UPDATE: I probably meant a regular commonplace fire of the usual temperature. That should simplify the answer.
 A: Broadly speaking, fire is a fast exothermic oxidation reaction. The flame is composed of hot, glowing gases, much like a metal that is heated sufficiently that it begins to glow. The atoms in the flame are a vapor, which is why it has the characteristic wispy quality we associate with fire, as opposed to the more rigid structure we associate with hot metal.
Now, to be fair, it is possible for a fire to burn sufficiently hot that it can ionize atoms. However, when we talk about common examples of fire, such as a candle flame, a campfire, or something of that kind, we are not dealing with anything sufficiently energetic to ionize atoms. So, when it comes to using something as an example of a plasma for kids, I'm afraid fire wouldn't be an accurate choice.
A: Firstly, 'Fire', according to numerous comments and answers [here][1] is a 'process', in which case, the answer to the question will be 'no', since plasma is a state of matter.
It would be unfair to leave it there by blaming the semantics, and given the abundant references to 'flame' region, I am going to assume that that is what the question meant to ask. I am also assuming that proving a candle flame constitutes plasma is enough to sufficiently answer the question.
From some papers (a quick google search gave me [2,3]) that flames have ionised content and that they are electrically conductive.
My suspicion was that not all flames are conductive, but [3] includes the statement:

It has been known for a long time that flames possess a high electrical conductivity and can be distorted by an electric field.

Sources [4] and [5], and numerous other sources, including a video on YouTube [6] claim that a candle flame is ionised and that's what causes the flame to be affected by electric field.
Now is it plasma?
The 'Plasma Coalition', which is a coalition of many reputed institutes around the world [7], says that ionisation alone is not enough, but enough atoms have to be ionized to significantly affect the electrical characteristics of the gas, in order for it to be called plasma. In one of its documents [8], it expands on this description in great detail.
It actually has a paper dedicated to this question, [8], which says that some flames contain plasma, whilst others don't. It expand further in sufficient detail, claiming that the answer depends on the region, what's being burned, the temperature, etc. 
It also acknowledges that the current knowledge about flames is quite limited to conclusively ascertain the charged particle densities at a particle location in the flame, as of 2008.
A wide variety of sources that claim that a flame (like a candle flame) is plasma is referring to the fact that it is ionised. 
Francis F Chen's book [10] includes an exercise on page 12 that connotes a typical flame being plasma. This claim is repeated in [4] and [5] (refers to candle flame).
My Conclusion
I understand that the Plasma coalition paper [8] says that the temperature of a candle is too low for much ionisation to occur, but technically, the experiments cited above [2,4,6], demonstrating the significant effect of flames in an electric field, coupled with the theoretical predictions [3,10] seem to imply that the flame is indeed a plasma. Even by the condition stated by the Plasma Coalition [11] itself!
I found it interesting that an old paper [3] proposes to explain the excessive amounts of ions formed in hydrocarbon flames by suggesting that it is in part due cumulative excitation or chemi-ionisation. I do not know if it is still relevant today. 
$\ \ $  [1] Is fire matter or energy?, Physics Stack Exchange.
$\ \ $  [2] Electrical Properties of Flames: Burner Flames in Longitudinal Electric Fields. Hartwell F. Calcot and Robert N. Pease. Ind. Eng. Chem. 43 no. 12, pp 2726–2731 (1951).
$\ \ $  [3] Mechanisms for the formation of ions in flames. H.F. Calcote. Combust. Flame. 1 no. 4, pp. 385–403 (1957).
$\ \ $  [4] Waves in Dusty Space Plasmas. Frank Verheest (Kluwer Academic, 2000, The Netherlands).
$\ \ $  [5] Sun, Earth and Sky. Kenneth R. Lang (Springer, 2006, Berlin).
$\ \ $  [6] What's in a candle flame, Veritasium YouTube Channel.
$\ \ $  [7] About the Coalition for Plasma Science.
$\ \ $  [8] About Plasmas. Coalition of Plasma Science, 2008.
$\ \ $  [9] Plasma State of Matter. Lecture notes for PX384 Electrodynamics at Warwick University, chapter IV. Erwin Verwichte, 2013.
$\ $  [10] Introduction to Plasma Physics and Controlled Fusion. Francis Chen. Available here for the moment.
$\ $  [11] What is Plasma?. Coalition for Plasma Science, 2000.
A: Back of the envelope calculation:
The Saha equation for a Hydrogen plasma says
$$\frac{N_i^2}{N_H} = V \left(\frac{2 \pi m_e k_b T}{h^2}\right)^{3/2} \exp\left(\frac{-R}{k_b T}\right)$$
where $N_i$ is the number of ions, $N_H$ is the number of Hydrogen atoms, $V$ is the volume of the plasma, and $R$ is the Hydrogen ionization energy (13.6eV).
Defining the degree of ionization $\xi = N_i / N_0$, where $N_0 = N_i + N_H$ is the total number of atoms in the system, this can be written
$$\frac{\xi^2}{1-\xi} = \frac{V}{N_0} \left(\frac{2 \pi m_e k_b T}{h^2}\right)^{3/2} \exp\left(\frac{-R}{k_b T}\right)$$
A candle burns at 1000 Celsius, and the flame has a volume of around 1cm^3, with probably 10^20 atoms in the flame. For simplicity, let's assume it's mainly Hydrogen in the flame (the ionization energy of other elements is of the same order of magnitude anyway, so we won't be far off). Then I make the right hand side of the equation (we'll call it $f$) to be around 10^-54. Then we can solve $\frac{\xi^2}{1-\xi} = f$ using the quadratic formula:
$$\xi = \frac{\sqrt{f^2 + 4f} - f}{2}$$
This gives us $\xi = 10^{-27}$: none of the particles in a candle flame are ionized (remember, we guessed there were only 10^20 particles). This makes perfect sense, because 1000C is only around 0.1eV, a good two orders of magnitude less than the ionization potential. The particle density is too low to make up for that.
If you think any of my approximations don't apply (personally, I'm not too sure about the particle density) then please correct me in a comment!
A: Nope. Fire is a thermal phenomenon, plasma is more of electrical.
What's plasma?
Plasma is the state when you strip off electrons/add electrons to a gas--so plasma consists of charged gas ions. It usually glows due to electron transitions and whatnot.
What's fire?
In a flame, you basically have hot soot/&c molecules flying up. Any hot material emits photons, which are usually in the infrared range for normal temperatures. At higher temperatures, they can go into the visible range.
One way to explain this is by blackbody radiation-- the soot must emit photons since it has a nonzero temperature.
What's actually going on is that the electrons are "thermally excited"--they have extra energy and are prone to making transitions. Transitions lead to absorption/emission of light, and this is what causes the color.
You can see that there aren't any ions involved in fire, so it's not plasma. But ionization will occur if you heat it to even higher temperatures, and it can become plasma.
A: Fire is a plasma. There are two kinds of plasmas: hot plasmas relevant to astrophysics or fusion are indeed a mixture of totally ionized gas. In cold plasmas ( northen lights, Neon tubes,flamme) the ionization degree is less than one but the mixture typically exhibit collective behaviour and a zoo of waves one do not encounter in gases. The most famous is the plasma oscillation and the Alfven wave but they are many others.
poorsod's calculus assume the ionization takes place between n=1 and n=infinity. In reality, the atoms are first excited by collisions, their electrons jump on higher n until their bounding energy is lower than thermal energy of free electrons. For 0.1 eV more than 99% of the atoms are ionized (I did work on a computer model to analyze this problem). Though the equilibrium Saha approach is known to be false (the electron distribution function is not Maxwellian), you can get a preety good idea of the problem if you split your neutral atoms population into atoms in the fundamental, n=2, n=3, etc.. and use Saha equation for each population. 
