If a person perspired, and then removed his shirt without touching his skin, he does not need to wash his hands, even though he touched the perspiration on his shirt. If the sweat contained a tangible, grimy substance, then he must wash his hands.
Hacham Bension Abba Shaul (Jerusalem, 1924-1998), ruled that one who touches the bottom of his feet while wearing socks must wash his hands. Since the bottom of the feet generally perspire profusely, touching the socks in that area necessitates washing. It is likely, however, that this would not apply nowadays, when we bathe regularly and change our socks daily.
If a person inserts his finger into his nostril or ear, he must wash his hands, just as he must after touching any normally-covered part of the body. If one's hand came in contact with nasal discharge outside the body, then, according to some Poskim – including the Gaon of Vilna (1720-1797) – he does not need to wash his hands. Others, however, rule stringently in this regard, and so it is preferable to wash one's hands. If, however, this occurs while one is learning Torah, he should simply wipe his hands on some material, such as a tissue, rather than take the time to leave and wash his hands.
Touching saliva does not necessitate washing. If one squeezes a pimple and it releases puss which comes in contact with the fingers, he must wash his hands.
If one's hand comes in contact with blood, he does not need to wash his hands, unless he touches a large quantity of blood, in which case washing is required.
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) writes that touching discharge from the eyes does not necessitate washing.
If one's hand came in contact with bodily waste – either urine or excrement – he must wash his hands. If one handled or touched a child's potty, or a used diaper, without touching the waste material itself, he does not need to wash his hands.