Even though diamonds have a very high melting point, in a pure oxygen environment the temperatures for flammability are fairly low (1320 C) in our "real world" atmospheres are but 20% oxygen, therefore the temperatures to burn them must be 1520 or so degrees.
Jewelers coat diamonds in boron when retipping prongs as the solder melting points are higher than the flash point of a diamond. Boron coating keeps from oxygen getting to the stone during the repair. So technically diamonds are NOT flammable but ARE combustible.
When a diamond does burn, there is no ash residue as a diamond is pure carbon and when burned (oxidized) it turns into carbon dioxide.
Just about everything will burn if exposed to a high enough temperature, but diamonds are certainly not considered flammable. This is because of the structure of diamond. Diamond is composed entirely of carbon, and each carbon atom is bonded two four other carbon atoms.
When something burns, the reaction is actually the combination of that substance with oxygen. So for diamond to burn, the carbon atoms must break their bonds to the other carbon atoms and form new bonds with oxygen.
Because of the interconnected and 3-D structure of diamond (the same reason it is one of the hardest substances on earth) would prevent this reaction from happening at any significant rate. Although if you broke the diamond down into the finest powder possible, it would definitely burn fast than if you had a single large diamond crystal.
So in a crematorium, it is possible depending on the crematorium as temps are 1300 to 1700 with residence times between 3 hours and 1 hour....