Is fire matter or energy?
Fire is neither. Fire is a process involving both. Fire is the energetic combination of various substances with oxygen to release light and heat. In a gas fire, such as might be found on a stove or in a heater, a light hydrocarbon such propane is broken down into components of hydrogen and carbon which unite with oxygen from the atmosphere to form water and carbon dioxide. In an ordinary wood fire, heat causes the wood to break down and release hydrocarbons which then burn as propane would.
I suggest that looking for an explanation for 'fire' may be the wrong approach. Science is a process where we try to explain what we perceive. In doing so, we may have to sacrifice (or at least temporarily suspend) a common perspective.
My presumption is that you are probably mostly interested in the visual manifestation of fire - the dancing flames. Explaining the existence of light and its color requires only correctly identifying the sources that emit the light (and showing how they got there). Those sources are known to be from excited electron and molecule de-excitement, plus some blackbody radiation from the larger particles, but this depends on the type of reaction. An ordinary camp fire or a candle, to the extent of my understanding, is a mix of the different sources.
The location of the flame above the reaction is due to convection, powered by the lower density in the hot gas (pushing it up). You also know that the flame follows the wind, which further substantiates the fact that the observed color is, in fact, afterglow kind of effects that come a short time after the main reaction. The dancing of the flame is the same type of movement you will see in any gas with different temperatures or subject to a heat source, which is a great breeding ground for turbulence.
Fire can not be contained and kept indefinitely. It is only there as long as the reactants are present. I mean, you have to have both the "fuel" and oxygen for the case of common fires. One claim that could be made is that fire is a gas that is actively emitting those colors. This may fit some people's definition for fire, but not others. It should again be stressed that a gas meeting this criteria must be actively loosing energy, meaning that it can not exist without constant energy input. Others may focus on the reaction itself, saying that fire is defined by the reaction, so your car uses fire. A favorite term of mine regarding nuclear power plants is the "nuclear fire", indicating that self-sustaining reactions in general are fire-like to some people.
This is how I slice it: I feel comfortable saying that fire is matter emitting energy in a self-sustaining reaction, or the energy that comes from such reactions, or the reaction itself. I see that giving answers of "matter releasing energy", "an energy flow", and "a process". I hope this helps clarify why physicists will often answer "neither" to the question.
In this Wiki article fire is described in physics terms, so I will not repeat it here. Wnoise is correct that fire is a process, so your question is more appropriate to "flames".
From the article you can see that flames are gases in various temperatures while molecular interactions increase the kinetic energy of the molecules and are also releasing radiation which goes into heat and light.
A flame has both matter and energy.
Matter is the molecules of the gas, Oxygen, Carbon, which combine and release radiation by combining, which radiation heats the whole gas/air mixture.
Energy in the flame is in the form of heat, i.e. kinetic energy of the molecules,plus in the form of electromagnetic radiation, photons, both infrared and visible.