Leviticus 3 gives the details of the peace offering as it pertained to God, but not as it pertained to the priests. Leviticus 7 adds:
And the priest shall burn the fat upon the altar: but the breast shall be Aaron's and his sons'. And the right shoulder shall ye give unto the priest for an heave offering of the sacrifices of your peace offerings. He among the sons of Aaron, that offereth the blood of the peace offerings, and the fat, shall have the right shoulder for his part.
Either the whole beast was burnt and the ashes taken outside the camp, or the fat parts and innards were burnt, the ashes dumped by the altar, and the remainder given to, and eaten by, the priests and their families.
This is why I celebrate the Communion with bread and wine (Matthew 26:26-29; Luke 22:17-20; 1 Corinthians 11:23-26). For me, the wine is the Father's portion: symbolising the life of the sacrifice as the necessary cost of forgiveness; but, the bread is mine: symbolising the body of the sacrifice, i.e. Jesus as the Word of God, by which I am sustained.
There is an expression "sweet aroma" that might be associated with the burning of the fat. But, as far as I can see, all offerings that are burnt on the altar are referred to as giving off a sweet aroma, which I take to be a reference to the pleasure God gets from the genuine desire of His people to deal with sin in the way He prescribed.
There is, however, in Psalm 66 a specific reference that directly connects "the sweet aroma" to "the fat", alone.
I will offer You burnt sacrifices of fat animals,
With the sweet aroma of rams;
I will offer bulls with goats.
-- Psalm 66:15 (NKJV)
The blood and fat were the Lord's, as were the innards, because those parts were not consumable by the priests.
The sacrificial system was not just about slaughtering animals. It was also about sustaining the people responsible for administering the system, who had been given no possession in the land.