Some flames do contain plasma and some flames do not. To properly answer this question, we really have to first strictly define what we mean by "plasma". A textbook definition of a plasma is an ionized gas. "Ionized gas" means that some electrons have been ripped completely off the atoms that make up the gas. The effectively-free electrons are negatively charged and the resulting ionized atoms end up positively charged. An "ion" is an atom with an unequal number of electrons and protons. This definition is a good starting point, but it is not exact enough. Every gas contains a few ions and freed electrons, and yet not every gas is a plasma. There must by some cutoff point where there are enough ions in the gas that it begins acting like a plasma.