A thread at Reddit addressed the question "If diamonds are made of just carbon, is it possible to get a diamond to catch fire?"
The embedded video answers the question by showing a diamond being burned (heated white-hot, then dropped in liquid oxygen).
Practical significance, for those without liquid oxygen at home and diamonds to burn?
If your house burns down with the family jewels inside, you can collect the pools of melted gold, but the diamonds will be gone in a puff of CO2. Cheaper, more attractive stones, such as cubic zirconia and synthetic ruby and sapphire, are made of refractory metal oxides that easily withstand the same heat. So it's actually mall trinkets, not diamonds, that are forever.