Who takes the Chelev from the animal - the sinner or the priest? What is the spiritual meaning of the burning of the Chelev?

All of the latter acts on the animal sacrifice, including the removing of the Chelev fats, were done by the Kohanim (priests). These fats were considered special delicacies by man, and therefore we sublime our urges for such pleasures, and give the "best parts" to be burned for God. Obviously God is perfect and doesn't need anything, especially such physical gifts, but it's good for our loving relationship with Him to restrain ourselves, and as if it were, "give the best parts" to our Beloved.