Is the cleansing fire of purgatory a real physical fire?
The short answer
The Church has not defined that. She has dogmatically taught only that there is purification after death, that it involves some pain, and that the souls can be helped by prayers, devotions and Masses. Whether it involves material fire is a matter of speculation.
What is defined
Only a few propositions about purgatory are dogma: there is purification after death; it involves some kind of pain; and those being purified can be helped by prayers, devotions and Masses.
What is speculative
Its duration, whether it is a place, whether punishments are actively inflicted, and whether it involves material fire are open questions.
Ludwig Ott's summary
Ott notes that the Latin Fathers, the scholastics and many modern theologians, on the basis of 1 Corinthians 3:15, assumed a physical fire, but the biblical basis for this is inadequate. Out of consideration for the separated Greeks, who reject the idea of a purifying fire, the councils speak only of purifying punishments, not of fire.