I'm looking into a suggestion I've seen mentioned occasionally, most recently in the comments for another question in Worldbuilding. The idea is that human respiration relies on having the correct partial pressure of oxygen, whereas a typical fire is less sensitive to pressure but needs a percentage of oxygen in the atmosphere. Therefore, if you reduce the %O2 and compensate by increasing the pressure, you can have a perfectly breathable atmosphere with a reduced fire risk.
I believe the first part is accurate: higher pressure increases the solubility of O2 in lung fluid, aiding absorption.
I'm not convinced by the second: higher pressure means a greater density of oxygen, which should allow a faster burn. A quick search confirms that at least some fires burn faster at higher pressures.
However, that doesn't rule out the possibility that the lung will "benefit" from increased pressure more than the fire.
To be specific, suppose I took a sample of Earth's atmosphere at sea level, replaced some of the oxygen with nitrogen, and increased the pressure so that it was safely breathable for humans. Would that atmosphere be as good for burning a wad of dry cloth as the typical sea-level atmosphere?