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You can burn a diamond, if you can get it hot enough. Coal starts to burn when it gets to 400°C, but diamond won't ignite until it reaches about 800°C. Mind you, it's a very expensive way of heating your house!

The reason for this difference is the way that the atoms of carbon are put together in the two materials. Coal is made from the remains of very old plants, and the carbon atoms are arranged in a rather untidy way, without any regular pattern. Think of the atoms as a pile of Lego bricks which you have just tipped out of the box. You can move them apart very easily, and join them to other bricks without much difficulty.

Now put the bricks together so that each one is joined to four other bricks around it. As you add more bricks, you get a very strong solid shape which is rigid and needs a lot of effort by you to pull it apart again. This is a 'Lego diamond'. In a real diamond, the bricks are actually carbon atoms, but the pattern is the same. Because of the regular way that the carbon atoms join together in a diamond, it is very hard - you can't push the atoms around easily.

When something burns, the atoms have to be separated from each other, and this takes a lot more energy for diamond than it does for the untidy heap of atoms in coal, which is why you have to get it much hotter to set fire to it.

Source: ScienceNet