Aside from cultural implications (correct me if I'm wrong but artificial diamonds are not so hard to obtain, right?) diamonds are pure carbon and are more unstable than graphite. So, why not use diamonds as fuel?
My approach: steel melts at 1500°C at least (and there are better steel that melt at higher temperatures). It is pretty high compared with diamond combustion temperature that is just 800°C and once you get to that point combustion will generate enough energy to burn diamond itself. So you can burn it in a steel piston.
Ok, I admit 800°C is pretty high compared with paper and wood (233–400°C) but it is a clean combustion (you obtain only CO2, assuming a constant oxygen input). From this point of view hydrogen is the only competitor but it rise to 500°C, that is not little.
So, what is the practical limitation in the use of diamonds as fossil fuel?