All organic matter seems to be inflammable when dried up. Petroleum, Alcohol, Wood, and Coal burns readily. Can I call it a general rule? Are there known exceptions? What makes them the exceptions?
Burning is a non-scientific term, it implies appearance of a luminous chemical reaction in a gaseous phase. It is commonly called a flame. It does not necessarily imply the presence of carbon or even oxygen. Boron and hydrogen compounds can burn in air, similarly, sodium burns in the presence of chlorine.
You can rather ask, can I oxidize carbon containing materials to carbon dioxide or carbon monoxide in the presence of oxygen? The general answer is yes. You already mentioned the examples of wood, sugar, coal etc. which are carbon rich materials. They will burn with a luminous flame. The carbon in inorganic materials such as refractory carbides, like silicon carbide, can also be oxidized that but only under extreme conditions.
IUPAC acknowledges that the "The boundaries between ‘organic’ and ‘inorganic’ compounds are blurred." So there is no need to create an arbitrary division.