The Torah says not to leave over the meat of the Pesach offering (Exodus 12:10), known as the prohibition of Nosar. Even though this prohibition doesn't get lashes, either because it doesn't have an associated action or because it's given over for positive rectification (לאו הניתק לעשה) (Makkos 4b), my (hopefully reasonable) premise is it's still prohibited.
This means that all edible meat has to be eaten. However, this makes the following statement of the gemarra regarding the gid hanashe (sciatic nerve) problematic:
Pesachim 83b, Chullin 91b:
שמנו מותר וישראל קדושים הם ונוהגין בו איסור
How could there be a custom not to eat the fat of the gid hanashe, if it's permitted to eat? If it's permitted to eat, then it should be prohibited not to eat it.
Similarly, the gemarra gives cases where Chazal prohibited certain parts of the animal. If it's actually permissible to eat, they're creating Nosar! Perhaps we have to come on to עקירת דבר מן התורה בשב ואל תעשה, but that won't help for a custom.