1. One may not leave a bathroom with drops of urine on his shoes.
2. There is a dispute about whether a person with excrement on his body may recite Shema.
3. Rav Papa says that certainly one may not recite Shema while excrement is leaving his body.
4. Even a Tahor Kohen may not enter the Azarah to serve in the Beis ha'Mikdash without first immersing in a Mikvah.
5. Washing one's hands and feet is the minimum requirement before performing an Avodah in the Beis ha'Mikdash.
A BIT MORE
1. This is because people will think that he is not healthy enough to have fathered children, and they will say that his children are not really his but are Mamzerim.
2. Rav Huna: He may. Rav Chisda: He may not.
3. In other words, if it is partially out of his body he may not recite Shema because it is particularly odorous. If it is on another spot on his body, it does not create such an odor, and he therefore may recite Shema (according to the lenient opinion above in #2).
4. There is a dispute about why this immersion is required. Rebbi Yehudah's opinion is that it is required mid'Rabanan in order to enable a person to remember -- before he enters the Beis ha'Mikdash -- whether he has become Tamei recently.
5. A Kohen Gadol or Kohen Hedyot who performs an Avodah without having washed his hands and feet that day disqualifies the Avodah that he performs.