Gemara

57b (Mishnah): If Reuven's animal fell into Levi's garden and benefited, Reuven pays the benefit.

Question: Why don't we say that Reuven is exempt because the fruits were merely 'Mavri'ach Ari (chased away a lion)?

Answer #1: Mavri'ach Ari is only when Levi intended to prevent damage. Here, he did not.

Answer #2: Mavri'ach Ari is only when Levi did not suffer a loss. Here, he lost.

115b (Mishnah): If a flooding river overcame the donkeys of Reuven and Levi, and Levi neglected his own (cheaper) donkey to save Reuven's, he is paid only like a worker (he is not compensated for the loss of his donkey).

If he said 'I will save yours and you will pay for mine', Reuven must pay him.

Bava Basra 41b: Once, Rav Yehudah obligated Rav Kahana to return property. Rav Kahana said that he will bring a letter proving that this is not the Halachah.

Rav Yehudah: Until you bring such a letter, you must return the property.

Kesuvos 107b (Mishnah - Chanan and R. Yochanan ben Zakai): If David went overseas and Reuven fed David's wife, he is not reimbursed;

The sons of Kohanim Gedolim and R. Dosa ben Harkinas say that he swears how much he spent, and he is reimbursed.

Rishonim

The Rif brings the Gemara on 57b-58a.

Rosh (6:6): The two answers go together. Mavri'ach Ari is when Levi intended to prevent damage, even if he suffered a loss, e.g. he paid Reuven's loan. Alternatively, it is even without Levi's intent, but only if Levi did not lose. If it is almost certain that Reuven would lose, or he saves him from the lion itself, he gets his wages. If Levi spent money, Reuven pays everything. A Shomer Sachar shepherd should gather shepherds to fight off a lion. His employer pays him back. A Shomer Chinam should try to gather them for free. This suggests that if a Shomer Chinam voluntarily gathered them for pay, this was praiseworthy, and he is paid back what he spent and for his toil. Since a Shomer Chinam need not hire people, he is like a stranger (who volunteered to fight off the lion). When Levi neglected his own donkey to save Reuven's, he is paid only like a worker, i.e. he is paid for his toil. He is not called a Mavri'ach Ari.

Hagahos Ashri (Kesuvos 13:8): Some say that just like one who fed another's wife is not reimbursed, if Levi or his wife was captured and Moshe redeemed him or her, Moshe lost his money. R. Baruch disagrees. Moshe loses only when he fed Levi's wife. R. Gershom says that if a Nochri stole Levi's house and Moshe bought it from him, Levi cannot take it back without paying Moshe what he benefited him. We say that friends would redeem his land only from a creditor, but not from a robber, who accepts only money, not words.

Mordechai (Bava Kama 58): If one redeemed captives even without their knowledge, e.g. he could not ask them whether he should redeem them, he collects from them what he paid to redeem them. This is not Mavri'ach Ari, for the damage was clear. Once, a captive had enough money to redeem himself, but said 'do not redeem me.' Rabbeinu Meir ruled that we redeem him against his will, just like we do not heed one who says 'do not bury me from my property.' If a man sold his daughter to be an Amah Ivriyah, we force him to redeem her if he has the money. All the more so we force him to redeem himself!

Yam Shel Shlomo (Bava Kama 6:15): Even though paying another's loan is Mavri'ach Ari, if a Nochri seized Yosef without reason and Moshe redeemed him, this is different and Moshe is paid. This is even if Yosef's relative says that he could have exempted him for free. If a captive told his family not to redeem him, we do not heed him, for it is a blemish to the family. However, we do not redeem captives for more than their value. It is optional to redeem himself or his wife for more than his value. This is if he was seized for money. If he was seized for Nefashos, he must redeem himself for any amount. Therefore, if another redeemed him, he must repay him.

Poskim

Rema (YD 252:12): If David redeemed Yosef from captivity, Yosef must pay him if he has the money. He is not Mavri'ach Ari.

Gra (18): See Tosfos Bava Kama 58a DH Iy (who proved that Mavri'ach Ari does not apply when he prevented a definite loss, like the Rosh).

Be'er Heitev (8): Mahari Veil (78) says that if one owes a Nochri or owes head-tax, this is Pidyon Shevuyim, and the Tzibur must pay for him. This is only if we know that he cannot afford it himself, and he does not regularly borrow from Nochrim so that others will redeem him. Maharam Lublin (15) says that if Nochrim claimed that one was Mezanah and seized him, if they seek to kill him, letter of the law we must redeem him. If there is no mortal hanger, letter of the law we are exempt, but we should redeem him to avoid Chilul Hash-m.

Rebuttal (Pischei Teshuvah 10): Maharam Lublin holds that letter of the law one must redeem him up to his value, even if there is no mortal danger. We are not exempt because he was delinquent. Even if one sold himself to Nochrim, we redeem him the first two times! Even if he really was Mezanah, he is not like a Mumar whom we do not redeem. One does not become a Mumar for one transgression, and all the more so since it was in private, and all the more so nowadays that many transgress this, and all the more so since there are no witnesses, and Nochrim are established to make false accusations. However, even when they seek to kill, letter of the law we do not redeem captives above their value. (The Posek) Beis Hillel disagrees.

Rema (ibid.): Yosef must pay immediately. He cannot say 'I will go to Beis Din, and accept their ruling.' If Yosef has a claim against David, he will take him to Beis Din afterwards. If not, no one would redeem someone else.

Beis Yosef (DH Motzasi): Mahari Veil (148,149) says that Ploni need not use his money to redeem Almoni if Almoni has money. Since he is obligated to save Almoni with Almoni's money, Ploni swears how much he spent, and receives it. This is like a shepherd who hired others to fight off a lion. Since it is clear that Almoni must pay, Ploni need not have a case with him. If Almoni has a claim against Ploni, he will take him to Beis Din. This is like the case of Rav Kahana. He was forced to pay immediately; we do not wait until he can bring a proof.

Shach (13): We collect even from minor orphans. Normally we do not, for we accept testimony only in front of the party, and it is as if a minor is not here. Here, we do not need testimony, for the Torah obligates people to save. Also, if we would not collect until they mature, no one would redeem an orphan with his own money (Mahari Veil 148).

Gra (19): See Rema CM 14:4.

Rema (CM 14:4): In monetary cases, one must pay immediately. If he can overturn the verdict, they will return the money.