Pesachim 13: Why Eliyahu Won't Show Up on Shabbat

Trying to make sense of the halakhah, and how it lines up with the disputants' positions. Which brings us to a discussion of terumah (and kodshim) and how to dispose of it on Erev Shabbat, when the first day of Pesach is Saturday night, and you can't burn on the 14th, because it's Shabbat. It's an open question for a time, which in turn raises the concern that Eliyahu might show up on Shabbat to adjudicate... Also - more on this question, and including the challenge of the Korban Todah and the accompanying bread. How current do the Sages treat this discussion, when they are, in fact, hundreds of years removed.