Gemara

(Mishnah): If one's clothes became soaked in water on the road, he may continue to walk in them.

Shabbos 147a - Mishnah: If one bathed in a cave or in the springs of Tiverya and wiped himself, even if he used 10 towels, he may not take them (with him, lest he squeeze them);

However, even 10 people may use the same towel to wipe their hands, feet and face and take the towel.

147b: The Reisha teaches that even though each towel is barely wet, since only one person wiped himself, he may not take them back lest he (forget, and) squeeze them;

The Seifa teaches that even though the towel is very wet, since there are many people, they may take it back. They will remind each other not to squeeze it.

Beraisa: One may wipe himself with a towel and leave it in the window. He may not give it to a bathhouse attendant, for they are suspected of this (squeezing, to give it to another bather);

R. Shimon says, one may wipe himself with one towel and take it back home.

Question (Abaye): What is the Halachah?

Answer (Rav Yosef): R. Shimon permits (here), and also Rebbi, Shmuel, and R. Yochanan:

Beraisa - Rebbi: We used to bring to R. Shimon (on Shabbos) oil and a towel, transferring them between Chatzeros, roofs and Karfifos, until the bathing spring.

Shmuel: One may wipe himself with a towel and take it back home.

R. Yochanan: One may wipe himself with a towel and take it back home.

Question: But R. Yochanan said that the Halachah follows a Stam Mishnah, and our Mishnah forbids returning towels, even if he used 10 towels!

Answer: (He holds that it is not a Stam Mishnah.) His text says 'these are the words of Ben Chachinai':

Mishnah: One may not go near the Kordima river.

This is lest he slip.

Rishonim

Tosfos (Shabbos 147b DH d'Tanya): A Talmid may not bathe with his Rebbi, unless the Rebbi needs him. Presumably, one Talmid sufficed, and he would return the towel himself, for R. Shimon permits this.

Rif and Rosh (Shabbos 62a-b and 22:9,12): If one's clothes became soaked in water on the road, he may continue to walk in them. The Halachah is, one may wipe himself with a towel and take it back home.

Ran (DH Masnisin and DH v'Ha): The Mishnah permits even 10 people to wipe their whole bodies with one towel and return it. The Mishnah did not say so, for it is uncommon (the towel would be too wet). No one forbids wiping with towels; the argument is only about returning them.

Question (Rashba, brought in Ran): We should be concerned for squeezing, like we are regarding a garment spread over a barrel. Rav Huna and Rava argue about if it is better to jump across a stream or walk around it, but no one permits walking through the water, lest his garments get soaked and he will squeeze them (113b)!

Answer: Since everyone who bathes wipes himself, to forbid wiping is to forbid bathing, and such a decree could not stand.

Rif (62b): One may not go near the Filuma river, for there is quicksand there in which one can sink. Some say that one who bathes there gets chilled, and it loosens the bowels.

Rambam (Hilchos Shabbos 21:29): One may not bathe in water that loosens the bowels, or in mud in which one sinks...for all of these are painful; "V'Korasa la'Shabbos Oneg".

Rambam (22:20): One who bathed may wipe himself with a towel and take it back home. We are not concerned lest he squeeze it. Similarly, if one's clothes became soaked in water on the road, he may continue to walk in them without concern.

Rosh (22:15): Rashi says that one may not go near the Kordima river lest he slip and fall into water, and come to squeeze.

Poskim

Shulchan Aruch (301:45): If one's clothes became soaked in water, he may continue to walk in them without concern lest he squeeze.

Mishnah Berurah (162): We do not forbid a person's clothing to him. Presumably, one may even put on soaked clothing to wear if he has nothing else to wear.

Kaf ha'Chayim (254): In Shabbos (Reish 65a) Rashi says that the clothes became wet in the rain. This implies that one may continue to wear them, but one may not put on soaked clothing. Some are stringent, for Tosfos (DH Mi) defends Rashi.

Rema (46): One may not move clothes that became soaked in water, lest he squeeze them. This is if he cares about the water.

Source (Mordechai Shabbos 442): The Mishnah forbids bringing back a towel used to dry oneself after bathing. We learn that one may not move anything soaked (if he is displeased when it is wet), lest he squeeze.

Question: The Halachah does not follow the Mishnah!

Answer #1 (Magen Avraham 58, as explained by Machatzis ha'Shekel): The Halachah does not follow the Mishnah because this would preclude bathing. In the days of the Talmud people could not abide by this, due to their habits and constitutions. Other than this necessary concession, the Halachah follows our Tana. In any case, 10 people may move it. Nowadays, people do not need to bathe every day, so it should be forbidden to dry oneself! It seems that since Chachamim did not decree then, it is permitted. One should not dry himself with something which he does not want to be wet.

Answer #2 (Gra DH v'Asur): The Halachah allows carrying a towel because one is not concerned if it gets wet. (The Rema does not discuss such things.)

Mishnah Berurah (171): It is forbidden only if they became soaked through much water.

Mishnah Berurah (172): One may move rags, because they are constantly wet (people do not care about this).

Kaf ha'Chayim (270): We are not concerned for other liquids that do not launder. L'Chatchilah one may soak garments in them, e.g. to strain red wine over it.

Rema: On Shabbos one may not go to a place where he can slip and fall into water, lest his clothes get soaked and he will come to squeeze.

Shulchan Aruch (48) One may wipe himself with a towel and take it with him. We are not concerned lest he squeeze.

Mishnah Berurah (175): According to the Magen Avraham we had to allow drying oneself, so we allowed even returning it. But once he puts it in its place at home, he should not move it. However, Eliyahu Rabah permits, because only a small amount of water is absorbed. The Gra permits because one is not concerned if a towel gets wet.