Gemara

107b (Mishnah - Chanan): If Reuven went overseas and Shimon fed Reuven's wife, he is not reimbursed;

Bnei Kohanim Gedolim say that he swears how much he spent, and he is reimbursed.

(Mishnah #1): If Levi is Mudar Hana'ah (may not benefit) from Moshe, Moshe may give the (half-)Shekel for Levi, or pay Levi's debt.

Question: Granted, he may give his Shekel, for this is a Mitzvah;

(Mishnah): When they take Shekalim (from the chamber) for Korbanos, they have in mind people whose Shekalim were lost, collected, or will be collected.

But why may Moshe pay Levi's debt? This saves Levi money!

Answer #1 (R. Oshaya): Mishnah #1 is like Chanan, who says that if Shimon fed Reuven's wife, Shimon threw away his money.

Answer #2 (Rava): Mishnah #1 is even like Chachamim (Bnei Kohanim Gedolim). The case is, the loan was on condition that the lender cannot demand payment.

Question: Granted, Rava did not answer like R. Oshaya, in order to establish the Mishnah even like Chachamim. Why didn't R. Oshaya answer like Rava?

Answer: Repaying such a loan is not like giving him money, but he spares him shame.

(R. Zeira):) The Halachah follows Chanan's two laws and ka'Yotzei Bo.

Nedarim 43a (Mishnah): If Levi is Mudar Hana'ah from Moshe, and Levi has nothing to eat, Moshe may tell this to a grocer. The grocer supplies Levi, and Moshe pays for it.

Rishonim

Rif: Mishnah #1 is like Chanan, who says that if Moshe fed Levi's wife, Moshe threw away his money. In the Yerushalmi, Rabah bar Mamal says that Chanan and Bnei Kohanim Gedolim also argue about one who paid Levi's debt without his knowledge. R. Yosi says that Bnei Kohanim Gedolim say that if Moshe fed Levi's wife, Levi must compensate him, for Levi did not want her to starve. Regarding a debt, he can say 'I would have appeased the creditor, and he would have pardoned it.'

Rambam (Hilchos Malveh 26:6): If Moshe paid Levi's loan document without Levi's knowledge, Levi need not pay him. Moshe wasted his money. Perhaps Levi would have appeased the creditor, and he would have pardoned the debt.

Rosh (13:8): R. Oshaya answered that Mishnah #1 is like Chanan, who says that Moshe lost his money. Since Levi need not pay one who pays Levi's creditor, it is not considered giving Hana'ah. It is like mere Gerama (causation). R. Tam and R. Chananel forbid paying other debts. It is permitted only to feed his wife. Even though this is Levi's obligation, the debt is not clear. If no one fed her, perhaps she would have economized and managed by herself. Even though Beis Din allotted to her food, she does not want people to consider her to be ravenous. Chanan agrees that one who paid a loan document gets paid back. This is proper Hana'ah; Levi had no way to avoid paying. R. Oshaya means that Mishnah #1 is like Chanan, and it discusses feeding a wife. Rava answers that the Mishnah is even like Chachamim (the lender waived the right to demand payment). The Riva proves from the Yerushalmi (cited above in the Rif) that Chanan and Chachamim argue about other debts. We learn from the Heter to give Levi's Shekel. R. Tam says that the Yerushalmi surely argues (with the Bavli), for it says that Bnei Kohanim Gedolim agree that Moshe may pay Levi's debt, and R. Oshaya said that only Chanan permits. The Yerushalmi permits paying his Shekel in any case, but the Bavli permits only if Levi gave it and it was lost. Likewise, the Bavli holds that they argue only about feeding a wife, and the Yerushalmi disagrees!

Rosh (10): Throughout Shas, we learn similar cases from each other. Here, it was necessary to say 'uka'Yotzei Bo (and similar cases)' lest we think that Chanan exempts only regarding feeding a wife, which was not a proper debt. Rather, the Halachah follows Chanan also regarding proper debts.

Rosh (Nedarim 4:2): Rava establishes the case of paying his debt to be a loan in which the lender cannot demand payment. The borrower pays when he wants.

Poskim

Shulchan Aruch (CM 128:1): If Moshe paid Levi's loan document without Levi's knowledge, Levi need not pay him. Moshe lost his money. (Some texts - perhaps, or we say that) Levi would have appeased the creditor, and he would have pardoned the debt.

Rema: This seems primary, unlike the opinion that obligates Levi to repay Moshe.

Beis Yosef (DH u'Mashma): The Tur infers that the Rosh holds like R. Tam, because the Rosh brought R. Tam's rebuttal of the Riva's proof. This is wrong. Pirush ha'Rosh in Nedarim (33b DH Tanan) is like Rashi (and the Rif), and in Kesuvos (13:10) the Rosh says that the Halachah follows Chanan and ka'Yotzei Bo, i.e. (they also argue about) other loans.

Rebuttal (Bach DH u'Mah she'Chosav): R. Tam says that Chanan agrees that one must repay one who paid his loan document, for there is no way to evade paying it. R. Tam agrees that Chanan exempts paying one who paid a Milveh Al Peh. R. Chananel holds like R. Tam, and he explains (Tosfos 109a DH Shnei) 'ka'Yotzei Bo' to mean similar cases. He must refer to Milvos Al Peh. The Rosh rules like Chanan and ka'Yotzei Bo, i.e. any Milveh Al Peh.

Defense (Shach 5): If R. Tam agrees that Chanan exempts paying one who paid a Milveh Al Peh, what forced R. Tam to explain that the Heter to pay Levi's debt refers to food? Perhaps it refers to a Milveh Al Peh! R. Chananel explains 'ka'Yotzei Bo' to mean similar cases, i.e. debts in which it was possible that he would not be liable, e.g. feeding his children. The Rosh in Bava Basra (3:14) says that an Arev Kablan who paid a debt is reimbursed only if he can prove that he was an Arev or the borrower admits.

Question: The Rosh in Nedarim (4:2) explains the Mishnah like Rava, that the lender waived his right to demand payment. This implies that if not, Moshe may not pay Levi's loan! Why did the Rosh omit R. Oshaya's answer, that our Mishnah is like Chanan?

Answer #1 (Beis Yosef YD 221 DH l'Kach): Even though Chanan exempts the borrower regarding any loan, Rava holds that Chanan forbids paying other loans for a Mudar Hana'ah. The Halachah follows Rava, for he is Basra.

Question: The Gemara said that Rava answered unlike R. Oshaya in order to establish the Mishnah even like Chachamim. It should have said that Rava holds that Chanan agrees that it is forbidden for a Mudar Hana'ah!

Answer (Beis Yosef DH v'Ein): The Gemara gave a better answer. Even if Rava would hold like R. Oshaya, who holds that Chanan permits for a Mudar Hana'ah, it is better to establish the Mishnah like everyone. Alternatively, the reason Rava disagrees with R. Oshaya (who says that Chanan permits for a Mudar Hana'ah) is because it is better to establish the Mishnah like everyone.

Answer #2 (to Question (iv) and defense of Tur - Drishah CM 128:2 DH d'Lo): Really, the Rosh holds like R. Tam. If not, what would be his source to say that Rava holds that even Chanan forbids other loans? This also explains why the Rosh brought R. Tam's opinion last. Pirush ha'Rosh in Nedarim explained like Rashi, for 'Chov' connotes a regular debt. However, l'Halachah he relies on what he wrote in this Pesakim. In Kesuvos (13:10), the Rosh says that the Halachah follows Chanan and 'ka'Yotzei Bo.' According to the Rif this refers to other loans. According to the Rosh, it teaches that if Moshe fed Levi's wife and Kalev paid Moshe, Levi need not pay Kalev. In Bava Kama (6:6), the Rosh brings from the Yerushalmi that the borrower can say 'I would have appeased him.' This is a mere answer to resolve the Sugyos. L'Halachah, the Rosh relies on what he wrote in Kesuvos and Nedarim, where the Gemara focused on this law. In a Teshuvah (73:9), the Rosh answers according to the (Rif and) Rambam, because the questioner held like the Rambam; the Rosh corrected the questioner's text of the Rambam.

SMA (3): I proved (Drishah 2 above) that the Rosh holds like R. Tam. The Halachah follows them, for Mahariyo (166), who is Basra, rules like them.

Defense (of Shulchan Aruch - Shach 5): The Yerushalmi is explicitly like the Shulchan Aruch, the Bavli connotes like them, and more than thirty Poskim rule like them: the Rif, Rambam, Ri, Riva, Mordechai, Ran... The opinions that argue are Batel to the majority. Mahariyo was unsure. He joins R. Tam (to other Shitos, to require repaying) only in a case when Moshe had money of Levi.