Fire, in this case, is two parts. The first is that fire is generated from an exothermic reaction, usually involving oxygen. The second is that what we perceive as heat is just faster moving molecules at least in general.
In addition to this, reactions this fun and exciting are generally not spontaneous unless one or more of the materials is highly reactive. A chunk of soduim/potassium in water is a fairly tame example and a good chemistry class introduction. Chlorine trifloride (yes, flourine is THAT good at what it does) is very nasty in is ability to burn things.
So where to go from there?
Your Question
Said character can't just magically create fire out of nowhere because DUH MAGIC, but has the ability to manipulate the speed of particles - speed them up to create heat or fire, slow them down to get rid of it.
What you are missing is that this technically isn't fire -- it's "just" kinetic energy. You can make things catch on fire if you heat it enough, but on its own, you aren't actually setting things on fire in a conventional layman's sense of the word.
Now it is entirely plausible that you can heat something up enough to catch on fire. Paper to roughly 233 Celsius comes to mind first thanks to Mr. Bradbury, but even something like hydrogen gas or wood burns if hot enough. Creating a line of heated fuel to react to burn is plausible. Where the energy comes to add heat from might be your plothole. And you will need to address that lest somebody go "Well why can't they just heat it to X degrees for Y effect?"
Probably the easier extension is to add a bit of heat to a large area to get that sort of heat haze effect that can be seen on really hot days. It might not have an everyday use, but it is something that isn't fire that can be done.
Another extension is that by manipulating the speed of matter directly, you can also remove speed from it, cooling it down. This will lead to the inverse, cryokinesis. Unlike with adding heat, your character's issue will be where the energy removed from the system goes.
Your major problem is that, unless this is a thing, why the correct way to kill people is liekly to burn them from within ... probably painfully. Making an inferno to trap them in while visually cool looking, would likely be highly inefficient for killing. Good for a delay though.
Your character's lack of immunity also make sense. You are concerned with a single break in reality without actually having the required secondary powers to handle things like this safely. The energy will want to escape the system and return to equilibirum.
From the lens of science, gasses are bound (roughly) by the Ideal Gas Law -- PV = nRT. So if you are increasing the temperature with your powers, then the pressure and/or the volume will rise to compensate. That is something that to consider when deciding limits -- when your control from an area is cut, then that hot air will do something to reach equilibrium with its surroundings. If one constrains a set volume of hot air, then only the pressure rises -- big boom once released all at once perhaps?
Another condieration: If you can manipulate individual atoms instead of molecules, then you could do something like move part of a water molecule so fast it splits into hydrogen and oxygen. Now split, it is susceptable to burning by again maniuplating the speed of them to incite combustion. This would work with methane and other hydrocarbons (Alcohol fire breath?). Doing this to table salt might be a bit nasty because the sodium in it will burn, and chlorine can do nasty things on its own. If this is possible then you have a plentiful source of fuel in water and the air around you. And by controlling where you pull molecules apart, one should be able to at least partially control where the fire goes.
