The Shulchan Aruch[2] rules that one need not remove tzitzit when entering a bathroom. In the Beit Yosef, he brings a proof from Menachot 43a. The gemara relates that Rav Yehuda would make a beracha every time he put on tzitzit but needed to do so only in the morning because he did not take them off all day. Presumably, he entered the bathroom and used the facilities during the course of the day, and thus the clear implication is that he did not take them off at all. Even though one must avoid bizuy mitzva (disgracing a mitzva), normal daily activity, such as using the facilities, does not fall into that category.[3]
However, one should remove a tallit before entering a bathroom.[4] This is due primarily to the extra respect given to an object that is used exclusively in tefilla. The Mishna Berura[5] allows urinating with a tallit on, and so one can be lenient upon entering an area that has only urinals.[6]
It is important to note that the requirement to remove the tallit is a matter of propriety and not an outright requirement; this fact has a common application. When one enters a bathroom, one is required to remove tefillin, which have a level of kedusha that tzitzit lacks. Since this causes a mandatory interruption in the performance of the mitzva of tefillin, one must make a beracha when he puts them back on.[7] However, the rule to remove a tallit is a lower level of requirement, and the interruption is not mandatory. Therefore, if one removes the tallit with the intention to put it back on after leaving the bathroom, he does not make a new beracha at that time.[8]