Gemara

Beraisa: It is forbidden to wash part of the body just like the entire body is forbidden. If one was dirty from mud or excrement, he may wash normally.

A woman may wash one hand to feed bread to her child. Feeding without washing makes the child vulnerable to a Ru'ach Ra'ah called Shivsa.

On Erev Yom Kipur R. Yehoshua ben Levi used to soak a cloth and wring it out and let it dry, to become like a dry Keli. On Yom Kipur he would put it on his eyes.

Rishonim

Rif, Rambam (Hilchos Shevisas Asor 3:1,2) and Rosh (8:2,3): It is forbidden to wash part of the body like the entire body. If one was dirty from mud or excrement, he may wash (Rambam - the dirty place) normally. A woman may wash one hand to feed bread to her child.

Rif and Rosh (8:7): On Erev Yom Kipur one may dry his face, hands and feet with a cloth and put it on his eyes on Yom Kipur. He may not soak it on Erev Yom Kipur, lest he squeeze it on Yom Kipur.

Rambam (Hilchos Shevisas Asor 3:1): It is forbidden to wash on Yom Kipur in hot or cold water.

Source (R. Mano'ach): Ta'anis (13a) forbids bathing for pleasure in hot water, but permits bathing in cold water. That is only regarding a Ta'anis Tzibur (due to lack of rain) which is only mid'Rabanan, but not regarding Yom Kipur. We learn the Isur to bathe on Yom Kipur from "Mayim Karim Al Nefesh Ayefah", which discusses cold water!

Rosh (8:3,7): R. Tam permits washing the hands in the morning even if one need not feed a child. If one's hands are dirty he may wash them; all the more so it is permitted to wash off the Ru'ach Ra'ah which is on the hands after sleeping at night, for it is dangerous to touch the mouth, eye or nostril before washing. Similarly, one may wash after eliminating or urinating and wiping the drops off his feet, for one may not pray afterwards until washing. One must wash only the fingers, not the entire hand. R. Tam says that the Heter to wash before feeding applies even to one who washed in the morning, unlike Rashi.

Beis Yosef (OC 613 DH u'Mah she'Chosav Hilkach): Nowadays we are not careful about feeding without washing because the Ru'ach Ra'ah is not frequent among us, just like we are not careful about Giluy.

Rosh (8:7): One whose mind is unsettled each day until he washes his face may wash his face.

Poskim

Shulchan Aruch (OC 613:1): If one was dirty from mud, excrement, or blood from his nose, he may wash. Only washing for pleasure is forbidden.

Taz (1): Just like one may not anoint to remove filth, one may not wash to remove sweat

Magen Avraham (614:1): One may wash to remove sweat. Anointing is forbidden, for it is an indulgence (since it suffices to wash).

Mishnah Berurah (1,2): If one was dirty in many places and it is difficult to wash just the dirty places, he may bathe his entire body at once, but without intent to enjoy it. One who is not an Istnis should be stringent not to wash to remove sweat, just like we do not anoint to remove filth.

Shulchan Aruch (ibid. 2): One may wash his hands in the morning and bless Al Netilas Yadayim, but he must wash only until the last knuckles of the fingers.

Kaf ha'Chayim (613:11 and 4:14): On Yom Kipur we rely on the Poskim who say that it suffices to wash the fingers. Alternatively, this suffices because Ru'ach Ra'ah is weak on Yom Kipur.

Pri Megadim (4:7): Washing the fingers suffices to remove Ru'ach Ra'ah. On other days we wash to the wrist to prepare for prayer.

Tosefes Yom ha'Kipurim (77b DH Emnam): The Rambam (like Rashi) does not permit to wash in the morning. He holds that R. Noson says that it is always dangerous not to do so, but Chachamim argue and say that it is dangerous only when feeding bread to a child.

Rema: He may not intend to enjoy washing.

Shulchan Aruch (ibid. 3): One may wash until the last knuckles after urinating and wiping the drops or after eliminating and cleaning himself, like one whose hands are dirty. If he wants to pray he may wash even if he did not clean himself.

Source (Gra): In the Mikdash a Kohen must be Tovel after eliminating and must wash his hands and feet after urinating (Mishnah 28a). This refers to Yom Kipur, like the previous clause (they brought the Kohen Gadol down to be Tovel...)

Beis Yosef (DH u'Mah she'Chosav v'Chen, DH u'Mihu): If one did not clean himself, there is no need to wash his hands, for they are clean. However, if he wants to pray he must wash because of "Hikon Likras Elokecha Yisrael", even after urinating.

Bach (DH u'Mah she'Chosav v'Chen, DH Aval): Even blessing "Asher Yotzar" is like prayer and one must wash before saying it. However, perhaps this is only if one cleaned himself or wiped the drops. It is best to clean oneself or wipe the drops, and then all agree that he must wash.

Pri Chodosh (DH v'Im): The Bach's solution is unnecessary. Since his sole intent is to prepare for prayer, it is permitted.

Magen Avraham (2): The Bach's solution is for the night. During the day, one prays constantly, so he may wash even if he did not clean himself or wipe drops.

Mishnah Berurah (5,6): One may wash before prayer only if he used the bathroom. Normally, one washes before every Tefilah even if his hands are clean. On Yom Kipur we are concerned for Poskim who say that this applies only to Shacharis. If one touched a sweaty part of the body, he must wash all the fingers of that hand. If one became dirty with mud, he must wash only the dirty area.

Rema: Similarly, a Kohen washes his hands before Birkas Kohanim even if his hands are Tehorim, like any washing not for pleasure.

Mishnah Berurah (7): He washes until the wrist.

Kaf ha'Chayim (15): The Rema did not distinguish, so presumably he allows only the fingers. We do not rebuke one who washes to the wrist, for he has authorities to rely on.

Shulchan Aruch (ibid. 4): An Istnis whose mind is unsettled until he washes his face may do so.

Bach (DH v'Chosav): The leniency for an Istnis is only if there is dirt on his face.

Mishnah Berurah (9): One may wet his finger to clean dirt from his face or secretions from the eye.