The Gemara says that Jewish people have already been assured that Elijah will come neither on a Friday nor on the eve of a Festival, due to the exertion.
If that's the case, can Elijah come on Shabbos or Yom Tov day?
The relevant source is Eruvin 43.
Essentially, the gemara there does not come to a conclusion as to whether Elijah can come on Shabbat or Yom Tov. It depends on whether there is a concept of techum shabbat in the air, which would prevent Elijah coming on Shabbat or Yom Tov.
Note that the gemara there assumes that Elijah will come one day before the Messiah, and therefore that the Messiah will not come on Shabbat or Yom Tov, since Elijah cannot come the day before, as noted in the question.