Why doesn't increasing the temperature of something like wood or paper set them on fire?

Before answering your question, it is important to understand how ignition of a solid material occurs. For fuels that contain hydrogen and carbon like paper, Ignition is a gas phase phenomenon . It is not the solid itself that ignites. Before a solid material can be ignited, it must be partially converted into a volatile (combustible) gas. This generally requires heat. It is the combustible gases at the surface of the solid that actually ignites, not the solid itself. The process of decomposing a solid to generate combustible gas is called pyrolysis. The ignitable gaseous products of pyrolysis need to be mixed with oxygen (air) in the proper ratio in order to be in what is called the flammable range.

Ignition of the gas/air mixture produced by heating the paper can occur in two ways. If you continue to increase the temperature of the mixture it may reach what is called its auto (self) ignition temperature and ignite. This would be the mechanism for the book on a heated pan. Alternatively, exposing it to a pilot ignition source, such as an external flame or arc, can also ignite the mixture. That would be your book surrounded by air and exposed to a flame. The temperature of the mixture at which this occurs is called the piloted ignition temperature, or flash ignition temperature. Generally speaking, the piloted (flash) ignition temperature is less than the auto (self) ignition temperature.

Returning to paper and your book, @StudyStudy mentioned Fahrenheit 451. That (233 C) happens to be the auto (self) ignition temperature of paper, made popular by the book of the same name. The original test used to determining that temperature comes from ASTM 1929 “Standard Test Method for Determining Ignition Temperatures of Plastics”, though the test is not restricted to plastics. The piloted (flash) ignition temperature using the ASTM test is about 177 C, which is less than the auto ignition temperature.

Now let’s consider your book on a pan. Since there is no flame or arc above the pan, any ignition that would occur would be auto (self) ignition. All other things considered, as noted above, auto ignition requires the gas be at higher temperatures than ignition involving a pilot source (flame or arc). What’s more, heating occurs on the bottom surface of book. Much of this heat is conducted away from the heated surface into the mass of the book by heat conduction as well as to the surrounding air by convection. Most of the gaseous products of pyrolysis that may be produced at the bottom are prevented from mixing with air, which is essential for ignition. The surrounding air above the pan dilutes those gaseous products that escape the bottom surface. What you are likely to get is a book with a charred bottom but no flaming ignition.

If the book is surrounded by air subjected to an external flame, the much higher flame temperature can quickly both cause pyrolysis (thermal decomposition) and ignition of the resulting vapors.

Hope this helps.

The pan starts out at room temperature, say 20°C. The gas flame starts out around 2000°C. The ignition temperature of paper is roughly 200°C.

So the flame can immediately ignite the paper, but the pan cannot immediately do so because it must heat up. The pan can ignite the paper if the flame can heat up the pan and the paper to the ignition temperature. However, depending on the pan, it is possible that the pan might not heat up to the paper ignition temperature if it loses heat quickly enough to the surrounding air.