As far as I know, most flammable liquids are volatile. Are there flammable liquids that are non-volatile?
The element caesium melts at 28.4°C which is slightly above "room temperature" but below human body temperature, so I think we can consider it a liquid in "normal" conditions. It is so flammable it readily self-ignites in the air, but if you put it in inert atmosphere you will see that it isn't particularly volatile.
Flammability and volatility of a liquid are related.
According to the Globally Harmonized System of Classification and Labelling of Chemicals (GHS), the term “flammable liquid” is defined as follows:
Flammable liquid means a liquid having a flash point of not more than 60 °C.
The European Agreement concerning the International Carriage of Dangerous Goods by Road (ADR) uses a very similar definition.
(…) have a flash-point of not more than 60 °C (…).
(Note that the definitions in some older national regulations and standards may refer to different temperatures.)
The flash point is the lowest temperature of the liquid at which application of an ignition source causes the vapour of the substance to ignite. (The flash point is an experimental value and not a constant physical-chemical property of the liquid. The exact definition and the resulting value depend on the used experimental method.) Thus, actually the vapour above the liquid (i.e. the mixture of vapour and air) and not the liquid itself is ignited. Therefore, for the liquid to be considered flammable, there must be a significant concentration of vapour in the air above the liquid at the given temperature.
If the volatility of the liquid is too low, it cannot be a flammable liquid. For example, biodiesel (flash point: 180 °C) and typical vegetable oils are not flammable liquids although these liquids can be burned.
Most flammable liquids are flammable precisely because they are volatile (and combustible at that), so that they easily form vapors which easily catch fire. That being said, various oils are flammable (albeit less so than gasoline) and at the same time not very volatile.
Flammability and volatility are unrelated
Whether things burn or not (i.e. whether they are flammable) is unrelated to whether they are volatile. Volatility only determines whether things are easy to set alight.
Consider coal, for example. This is a non volatile solid but it is flammable though hard to set on fire. Long chain extracts from crude oil (tars and bitumen) are (technically) liquids but not at all volatile. They burn, though not easily (which is why some motorway accidents involving fires take so much time to repair). A range of other hydrocarbon fractions from crude oil are more liquid but not very volatile as are many vegetable oils (used in cooking because they are not volatile even at 200 °C but clearly flammable as you can see if splashed into a flame).
And there are plenty more examples. The confusion arises because many very volatile liquids are volatile enough that their vapours are flammable at room temperature. There is a good reason why you don't use bunsen burners anywhere near ether (bp ~40 °C): the vapour will ignite and flash back to the vessel even if the flame is far away. But many non-volatile things will also burn.
To tell the truth, "Flammable" means substance with easy ignitable vapors. But topicstater asked about "easy ignitable"- wider set of substances. Among weakly volatile ones I can add: 1)potassium-sodium eutectic alloy,liquid at ambient temperature.
2)white phosphorus - melts at ~40C,selfignites in the air.
3)thermally unstable chemicals like peroxides/nitrocompounds/some metal organic substances. Di-tert butyl peroxide (CH3)3COOC(CH3)3.
tert butyl hydroperoxide (CH3)3COOH ( https://pubchem.ncbi.nlm.nih.gov/compound/tert-butyl_hydroperoxide#section=Computed-Properties )
They easily react with oxygen of air (with or without heating) with subsequent selfheating to ignition. Or violently decompose, selfheat and finally burn.
Cooking oil is flammable, but doesn't evaporate or evaporates very slowly over months or years. http://www.oliveoilsource.com/asktheexpert/does-cooking-oil-evaporate It's flammable because if you heat it up enough, it will start to burn. Once it starts to burn, the heat will increase causing it to burn faster.