It depends on what time period.
In the Homeric period, sacrifice was a big deal. For example, in the Iliad you can read of "hecatombs" (hundreds of heads) being sacrificed at a time. Most sacrifices in old Greece did not happen at temples. You could (and were expected to) sacrifice everywhere. Normally only the less desirable parts were burned, such as the entrails, the rest being eaten. If you did your own sacrifice, you ate the cow. If you sacrificed at a temple, the priests got the cow. In case of a REALLY big deal, you could do a "holocaust" (burn the whole thing) in which case you burned the whole animal, which was expensive both in terms of fuel and time and of course you lost the whole cow (or sheep).
In later "classical" times, Greek sacrifice became more ceremonial. Often just a token chunk of the animal was burned and this was usually done at a temple. When the gods just got the token chunk, often it was one of the choice pieces of the animal (I guess one piece of filet mignon is better than 20 pounds of entrails).
The ashes were dumped into a pit and in some temples and oracles the ash pit could get very big. The ash pit at the oracle of Delphi was known as the "Omphalos", meaning "navel", insinuating it was the center point of the world.