Question: Ok, so I'm willing to bet at least one person has looked at the title of this question and thought to themselves "What?", so I'll try and explain this as thoroughly as I can.

Most incarnations of Pyrokinesis I've seen in fiction often deal with either telekinetically starting fires by exciting the flammable properties within the objects around them, or just straight up creating and releasing fire from within a person's body (the science behind this particular incarnation of the ability often varies from each piece of work). Some pyrokinetic people in fiction however, are capable of moving and shaping fires that are around them, sorta like how Aerokinesis lets you move air/wind and apply a shape to it.

Since fire is nothing but the result of a flammable material/substance getting exposed to oxygen and heat, this got me thinking. What are pyrokinetic people actually moving when they use their powers? Calling it fire seems inaccurate seeing as how fire itself lacks weight, cannot exist without certain requirements, and is treated less like a 'thing' but more so a chemical reaction. My understanding on science is how you say "Painfully mediocre", so I wonder if any of you guys and girls could help me out?

Answer: 
**Moving a match**

How do you normally move fire? You might shape the fire with funnels or wind, but the starting point is always the same. The flammable material. If you want to move the fire of a match, you move the match. Moving the match moves the flammable material, which in turn moves the origin point of the flame.

With telekinesis you want to move the flammable material as well, be it oil fumes or burning wood. Shaping of flame can be done afterwards with wind and funnels, which is basically pressure the flame cannot overcome. Telekinesis can so this as well while mindful of the limitations of the flame, like reach, starting point and duration.
This is the type of insightful deliberation that totally kills fantasy. There is a reason why it is called 'fantasy'. It just has to be something you do not think about or subject to intense scrutiny. Better to just leave it as, well, fantasy.

You are absolutely correct in bringing our attention to the reality that 'fire' is a very nebulous term. It is a concept. Really, no one has ever been injured by 'fire', they are injured by either the intense heat, or the toxic components of combustion, or the combustion (chemical reaction) itself. Firefighters are trained to and learn to ignore the 'fire', but concentrate on the individual very real dangers. That 'flame' will not injure you, but the heat, the smoke, the weakened floor, and lack of oxygen will. Any 'manipulation of fire' would, in essence, be the manipulation of one or more of these factors, not the 'fire' itself.

As you point out, what we call 'fire' is really just a perceptual, conceptual thing, not a physical thing. It only occurs in our mind. What exists in reality is the heat, the radiation, the products of combustion, and the results of combustion (the destruction of the fuel as it changes its chemistry). These things we conceptually all lump together and label as something called 'fire', and then put an emotional tag of 'bad and dangerous' to it. Once labeled and tagged, our minds have the distinct ability to handle it as a concept, and respond to the emotional tag. Thereafter the concept can be manipulated and altered, without needing to reference the underlying reality of the 'things' that actually make up the concept, and we continue to apply the emotional tag to the manipulated concept. We treat the concept, and the emotional tag, as a separate entity from all of its parts. The 'whole' still exists even if none of its parts continue to exist. We become afraid of the tag, the label, not the reality. Long after there is no danger from the reality of the components of this concept we call 'fire', we are still in fear of the concept called 'fire'. The word itself takes on a reality of its own.

It's like talking about the hazards of 'falling'. There is no such thing as ***a fall***. Everything is 'falling'. There is absolutely no danger in a fall, per say. It is the sudden stop at the end that creates the damage. Yet we still fear the fall, not the sudden stop at the end. Even in amusement park rides, when the danger of any sudden stop is removed, we still fear the fall.

Like saying 'I don't like vegetables', as if the label were a real thing. As the commercial infers, say 'fries are vegetables', and suddenly you do not like them.

Fantasy is really all about these labels and tags that our mind forms and creates, and our emotional reaction to these tags, not the reality behind them.

So render unto fantasy what is fantasy, and onto reality what is reality. Never conflict the two. Humans tend to like their emotional tags, thank you very much.