Eucalpytus is a hardwood infact it makes the best wood for cooking outdoor BBQs. The wood makes the best coals and lasts for hours and is practically smokeless.
There are two species of Eucalyptus that are so hard (i.e Iron Bark and Turpentine) you need an auxilliary fuel like coal to keep it burning.
Eucalytus has been used for cooking in Australia since its beginning and there are no reports of anyone ever becoming sick from using it.
Eucalyptus oils are in the leaves only and there is no toxic fumes associated with the wood when burned other than a little smoke.
If you are kucky enough to have a Eucalyptus tree in your neighbourhood and you can get some dry branches then try burning them in an open BBQ, you wont be dissapointed.
Eucalyptus has to be dry before you burn it, there is no sap produced when burned however there may water expelled at either end if the wood has been soaked with water and will burn a little slower.