Take a planet with an atmosphere somewhat similar to Earth's but denser, say 9-10bar incl 2-3bar O2, rest mostly N2 with some CO2 (0.1-0.3bar?) Temperature likewise similar to Earth, in the liquid water range. (Yes, I realize a denser atmosphere and in particular more CO2 means stronger greenhouse effect. This planet presumably receives less sunlight than Earth.) In discussions of atmosphere density and composition (IE, What's the useful range of atmospheric oxygen content for fire? or Fire on different oxygen density ), people often assert that higher O2 partial pressure makes fires much easier to start and/or spread, to the extent that carbon-based life can't survive or that such an atmosphere won't persist in nature because it's so reactive. However, such people tend to refer to situations like 0.5bar O2 in a 1bar atmosphere, or the Apollo 1 fire in a 0.3bar pure O2 atmosphere. My intuition says that also having more non-flammable components in the atmosphere should counter this effect, though. Sometimes (IE, What effects would an atmosphere with the following composition and pressure have on humans? on the environment? what color would the sky be? ) someone points out that a denser atmosphere can absorb, and carry away, more heat, making it harder to maintain the ignition temperature and harder for fires to spread. Given that convection dominates over conduction and radiation in terms of heat transfer in our atmosphere (that's why we use vacuum bottles for insulation), to what extent is this true? If conduction dominated, it's easy to see that denser air would carry away more heat, but does convection behave similarly? And most importantly, which effect dominates: increased flammability from higher O2 or reduced flammability from higher N2 and CO2? In general, is an atmosphere like I described unstable, and does it pose a serious threat to a biosphere through flammability? (Yes, I know most Earth life can't adapt to that atmosphere, but I'm not talking Earth life exactly. More on the biological aspects in another question.)
The flame will grow brighter and hotter. Partial pressure is the dominant parameter here. http://www.spaceref.com/news/viewsr.html?pid=4051
Video of a candle in 2-atmosphere chamber: https://www.youtube.com/watch?v=8WUdUE3qZFg
The increasing pressure will influence the laminar/turbulent forking points, and will thereby influence the form of the flame.
(I fully admit this answer is not "hard science", but it should serve as a stopgap until a real answer arrives)
Rate of combustion scales directly as partial pressure of Oxygen increases. However, due to heat concentration the scaling is not linear but a muddled higher-order relation, dependent of many factors including exposed surface area, radian heat ignition, diffusion rates, and many,many more.
It also deceases, somewhat, as partial pressure (actually volume mass) of buffer gases increase.
A nice classic example:
A stoichiometric mixture of Acetelyne into Air at sealevel pressure only burn(deflagrates).
The exact same mixture at 2 bar(or higher) of pressure detonates, i.e. burns with a flame front exceeding speed of sound in itself.
Simply put, in your specified atmosphere of 7 bar Inert + 3 bar O2, ignition will be easy, and fire burning rates catastrophically high. Well above 10 times the normal rate on Earth, although final flame temperatures will be only very slightly higher, due to the decreased ratio of buffer gas to Oxygen.
Some detail in articles such as:
Effects of Pressure and Oxygen Concentration on the Combustion of Different Coals
1974 report on fire hazards in oxygen-enriched atmospheres