Hi Spencer,
This is a good question, but I'm afraid it does not have a simple answer. That is what makes it such a good question!
The scientific defintition of a plasma is a quasi-neutral gas of charged and neutral particles which exhibits collective behavior. The "quasi-neutral" part means there are equal numbers of positive (usually ions) and negative (usually electrons) charges so that when viewed from the proper distance, the whole plasma appears to be electrically neutral. The "collective behavior" part means that the motions of particles in the plasma depend not only on local conditions, but also on the state of the plasma in remote regions. In an ordinary gas, collisions between neutral particles and charged particles occur so frequently, that the gas behaves like a normal fluid. In a plasma, these collisions occur less often and their motions are more strongly affected by electromagnetic forces. Scientists use a parameter called the Debye length, which depends upon the density and temperature to help determine whether or not something is a plasma. Nearly every gas will probably have some ionized particles (positive ions and negative electrons) in it, but that does not necessarily make it a plasma.
These conditions mean that some flames may be classified as plasmas, but others cannot be. It all depends on the material being burned and the temperature at which it burns. The flame on a wax candle burns at about 1500 degrees Celcius, which is too low for very much ionization, so a candle flame is not a plasma, it is just a gas. A burning mixture of acetylene and oxygen may be a plasma under some conditions, as its temperature can exceed 3000 degrees Celcius.
Here is a good pdf file describing the conditions for calling something a plasma and how this applies to flames.
http://www.plasmacoalitio...a_writeups/flame.pdf
Kris