Some inorganic compounds are combustible, believe it or not! They're just not very useful as day-to-day fuel, most of the time. All a combustion reaction is, is a reaction between a fuel (like gasoline) and an oxidizer (like oxygen in air) that generates heat. Magnesium metal, for instance, will burn very bright and hot if you can manage to light it! (video: https://www.youtube.com/watch?v=Wbfjyp9ZyhM)
A lot of inorganic compounds, like you said, are not combustible, and that is for one of two reasons:
1: They are already oxidized. A lot of ceramics (like glass) are made of metal oxides, which is to say metals that have already reacted with an oxidizer (like oxygen, chlorine, etc.). You can think of it like it's already been "burnt up". Sodium metal may react with a lot of things, but once it's been oxidized by chlorine to make sodium chloride, it's not going anywhere.
2: The inorganic compound isn't volatile. So it won't evaporate enough to let new unreacted material get exposed to the oxidizer. Oftentimes the surface will get a layer of oxidized material (and more like case 1), which keeps the rest of the material from making contact with the oxidizer.
A good example of this is aluminum. Take a torch to a block of aluminum, and not much happens (it might melt, bit not enough evaporates to make a reaction visible). Bu if you make it a fine powder (so the many small particles makes it behave more like a gas) you get a much more exciting reaction: https://www.youtube.com/watch?v=FhG2e6mpdEY
Hope that helps!