1. According to Rabbi Eliezer, an asham taluy can be donated at any time, even if there is no reason to think that the person sinned.
2. Baba ben Buta suggests that an asham taluy cannot be donated unless there is a possibility that the person sinned.
Effectively, the difference between these opinions is whether an asham taluy should be viewed as an obligatory sacrifice, or a sacrifice that can be donated. Rav Ovadiah mi-Bartenura points out that aside from the impact that this argument has on Temple practice, there is also a practical implication in the modern world. According to the Gemara in Hullin (daf 41b), were someone to slaughter an animal today – when the Temple is destroyed and there is no altar to bring sacrifices – and announce that his intention was for a sacrifice, we ignore his statement and the animal is kosher and can be eaten. This is only true, however, if his statement was that the animal was slaughtered for an obligatory sacrifice, when it is obvious that his statement was meaningless. If he said that it was slaughtered for a sacrifice that can be donated then we forbid the animal since it appears as if he has slaughtered kodashim ba-hutz – sanctified animals outside of the Temple precincts.
One of the examples that the Mishnah in Hullin offers as a sacrifice that can be donated is an asham taluy – which is only true according to the opinion of Rabbi Eliezer in our Mishnah.