If one is wearing Talit and Tefillin and suddenly becomes violently ill, what should one do? One cannot enter a bathroom with them on; however, one also runs the risk of fouling himself, or regurgitating in a place that is not proper.
I am going off of the Shulhan Arukh HaRav on this one. Not my normal source for halakhic information, but considering we are talking about a case of in extremis there is what to be lenient. The relevant portion is O"H Simian 43. Obviously one should do everything possible to keep from entering into a toilet with tefilin on, however that is not in the case above entirely possible.
Going with the common custom of pulling one's shirt sleeve, down over the tefillin, and then putting on one's jacket atop that, we have the double covering over the tefilin shel yad that the Shulhan Arukh HaRav is concerned about. We still have the wrapping around the hand, the tefilin shel rosh and the talit to worry about, as technically none of those can enter the toilet.
So my solution thus far, is to remove the talit and tefilin shel rosh(something that can be done relatively quickly). As well as to unwind the straps around one's hand and wind them further up the arm so that they will be covered by the sleeve(something that from recent experience I know can be done on the run). Thus attaining the double covering and permitting one to enter the toilet(in a situation of in extremis) without transgressing an issur.
As an additional leniency on which to rely both the Hazan Ish(can't remember the location) and Rav Ovadiah Yosef(Halikhot Olam Parashat Vayetze) rule that our toilets today do not have the strict din of a beit hakise spoken about in earlier law codes on account of them flushing and thus not always having a bad smell. Rav Ovadia compares it to relieving oneself in a field.