Must a person stop learning if he develops the need to relieve himself? 

We have previously learned that if a person began Shemoneh Esrei or Kriyas Shema without the urge to relieve himself, he does not have to stop if the need develops afterwards. In fact, it is forbidden to stop during Shemoneh Esrei. The poskim extend this halacha to other parts of learning and davening. Thus, if someone answered Borchu during Kriyas HaTorah, he may remain in place even if he develops the need to relieve himself during laining. He may respond to the half-Kaddish after laining but must stop before the Maftir aliyah, since Maftir is a different aspect of laining. Some hold that even if one began to learn on his own before needing the bathroom, he may complete the entire topic which he is studying if the need arises while he is learning. One certainly does not need to be stringent if stopping would cause him not to return to his learning.

סצ"ב ס"ב ביה"ל ד"ה קורא ביאורים ומוספים דרשו 22

Should one stop in middle of Shemoneh Esrei to use the bathroom?

Rema rules that if one develops a strong need to relieve himself while davening -- to the extent that delaying would cause him to transgress the prohibition of bal tishaktzu -- he must stop and relieve himself. Other Achronim disagree and say that one should not stop in the middle of Shemoneh Esrei (assuming that he had no need when he started) even if a very strong urge develops. If this occurs during Kriyas Shema, however, all agree that he must stop. A third opinion holds that one may choose to stop but is not obligated to do so. If he does stop, he may walk within four amos of another person who is davening.

ס"ב ס"ק י"א ביה"ל ד"ה אבל ביאורים ומוספים דרשו 19

Under what circumstances would one have to repeat Shemoneh Esrei?

If one is unable to complete Shemoneh Esrei due to his need to relieve himself, he must stop and leave the shul to attend to his needs. When he is done, it is unclear whether he must return to his place in shul to complete Shemoneh Esrei. If the interruption lasted the amount of time it takes to complete Shemoneh Esrei, he must start again. Regarding this halacha, the time which was spent walking to the bathroom and using the bathroom counts towards the length of the break. The time it takes to return from the bathroom does not count in the calculation.

ס"ק י"א ביה"ל ד"ה יותר ביאורים ומוספים דרשו 21

• It is forbidden to daven or learn Torah if one has the need to relieve himself. Although this may mean that one has to forfeit tefilah b’tzibur, he may not daven in this compromised state. If one did daven, his tefilah is not valid.

• Aside from the prohibition against davening or learning when one has to relieve himself, there is a general prohibition against holding oneself back from using the restroom. This issur is known as bal tishaktzu. To transgress this prohibition, one must have a strong need to relieve himself. This issur is only mid’rabonon. Therefore, if one is giving a lecture and it will be embarrassing for him to interrupt it and tend to his needs, he may finish the class.

• If the need arises while he is in the middle of Kriyas Shema, he may choose to complete Shema but it is forbidden to continue on to Shemoneh Esrei. It is best to use the restroom between Shema and Emes V’yatziv. If he did not do so, he should go in the middle of the brocha but not at the end of the brocha, so as not to interrupt between geulah and tefilah. 

• How far does one have to travel to get water to wash for tefilah?

• Is one obligated to wash before davening if his hands are not dirty?

• What is the appropriate way to clean one’s hands when learning?