Fire is hot enough to ionize some molecules, but not enough to qualify it as a plasma. Generally, in any gas (even at room temperature) some of the particles will be ionized anyway, so that alone isn’t the defining factor. A better example of plasmas are electric arcs – like from arc welders or even if you have a Tesla coil handy.
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The fire would have to burn sufficiently hot enough to ionize atoms, though that alone isn’t enough to be considered plasma. Enough atoms have to be ionized to significantly affect the electrical characteristics of the gas.
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