The Mishnah taught (Chulin 119(f)) that R. Yehuda says that one may light it at a banquet but not in a house of mourning. R. Yosi permitted it in a house of mourning but not at a banquet. R. Meir prohibits both and R. Shimon permits both.

What's R. Yehuda's reasoning? In a banquet, since their clothing is clean, they are careful not to touch the candle (so there's no concern that the candle will be taken away from the Kohen). In a house of mourning, where their clothing isn't clean, they might touch and move the candle.

What's R. Yosi's reasoning? In a house of mourning, since their hearts are subdued, they won't move the candle. In a banquet, in their state of joy, they might move the candle.

What's R. Meir's reasoning? In a house of mourning, since their clothing isn't clean, they might move the candle. In a banquet, in their state of joy, they might move it.

What's R. Shimon's reasoning? In a house of mourning, they are subdued and won't move it. In a banquet, their clothing is clean, so they won't move it.

(D'Bei R. Yannai): The Halacha follows R. Shimon (who is lenient in both cases).

(R. Yaakov bar Acha citing R. Yashiya): The Halacha follows R. Shimon.

Question (R. Yosi Tzirania to R. Yirmiyah): Why is this ruling needed? We already know that when R. Meir disagrees with R. Shimon, the Halacha follows R. Shimon?! (See earlier in Perek 3 Halacha 1 - Chulin 71(s))

Answer (R. Yirmiyah): R. Yehuda and R. Yosi disagreed over it, and we have a rule that the Halacha follows the opinion that resolves the dispute. Here, R. Meir prohibits both and R. Shimon permits both; and it is R. Yehuda and R. Yosi who come to resolve - so we might have thought that the Halacha follows R. Yehuda and R. Yosi over R. Meir and R. Shimon. If that would be so, when R. Yehuda and R. Yosi disagree, the Halacha follows R. Yosi. Therefore, we needed to be told that rather, the Halacha follows R. Shimon.

Question: Is it permitted to use Tamei Terumah oil for lighting the Chanukah candles?

Answer (D'Bei R. Yannai): It is permitted. (The Mitzvah is to publicize the miracles, which is considered a public need, similar to lighting in synagogues and study halls.)

(R. Nisa): I didn't know my father (as he passed away when I was young). My mother told me that he said that it is only permitted to use Tamei Terumah for the Chanukah candles when there is no Chulin oil available.