What a ripper of a question! We have been asked some absolute beauties in our time, but this one is up there with the best. The staff here at the Koala Hospital have visions of koalas exploding in trees all over the Australian bush when temperatures get too high and the eucalyptus oil within their bodies and on their fur ignites.  We can just picture balls of flaming fur flying through the air with stunned smoking koalas landing on the ground with a thud, shaking the soot, ash and cinders out of their blackened fur and in a weak, dry crackling voice saying, "Stone the crows...its bloody hot today."
 
If this was a true fact, there wouldn't be any grey fluffy furry koalas in Australia, only black balls of smouldering smoky marsupials sitting in the forks of leafless crispy eucalyptus trees.

First we must note that koalas are NOT bears. They are not a placental mammal but a marsupial mammal with pouched young. Koalas have evolved alongside the Australian Eucalypts for thousands and thousands (probably millions) of years and have developed not only strategies to cope with the volatile oils in the eucalyptus leaf but also have developed strategies to cope with high temperatures.

No they most certainly do not ignite.  When temperatures in summer reach about 40 degree C (104F), koala behaviour dictates that they move to lower more cooler areas of the trees, they even move to trees that are not eucalypts, that may offer more shade.  Their fur has double thickness, which insulates them from both the cold and the heat.  Temperatures of 40-45 degrees would not be hot enough to cause ignition anyway.  It would be an extremely rare event for the temperature to go higher than 45 degrees (113 F) in areas where koalas are normally known to inhabit. On the western plains of Queensland and New South Wales where temperatures will go higher and the koala population is very scattered, the koalas behaviour again would dictate that they would seek cooler/shadier spots during the heat of the day.  They are not silly creatures and would not expose themselves to extremes.

Eucalypt trees DO NOT ignite at 110 degrees (Fahrenheit??).    Eucalypt oil does not ignite in the leaf due to the suns heat radiating on them.  If eucalypt oil was able to ignite at 110 degrees Fahrenheit then we would not have any eucalypt forests in Australia -- they would have all become extinct a long time ago.

The koalas have a large multi-lobed highly efficient liver and gut system which copes very well in dealing with and eliminating the toxic oils in the eucalypt leaf.

There are no recorded cases of spontaneous combustion in koalas. If they were able to ignite, the koala population probably would have become extinct millenia ago.  Thank goodness for natural selection and evolutionary strategies of nature!