Can I use diamonds as fossil fuel?
Diamonds are expensive. Really expensive. Even "cheap" synthetic diamonds are orders of magnitude more expensive than conventional fossil fuel. By using them as fuel, you will increase demand, thus increasing their price even more. And synthetic diamonds have to be made somehow, and you need energy for that.
Diamonds burn, but they don't burn well. For example, look at this video of diamonds burning in pure oxygen. The burning sustains for about a minute after the heat source is removed, but then it stops. So then, you need to apply a constant source of heat, and this requires energy. You will probably have to put in more energy than you get from the combustion of diamonds. Kinetically, diamonds just don't burn as well as other things with carbon (like fuel or wood or coal).
One of your arguments is that diamonds burn "cleanly" (let's not forget that CO2 is a pollutant as well). Burning of natural gas emits only CO2 and H2O, so it's a much better alternative when pollution is a consideration.
Diamonds are so much more useful for other things. Don't waste diamonds on energy production!