The Gemoro (Eruvin 43a) assumes that אליהו הנביא
cannot appear on Shabbos or Yom Tov,
because of the Issur of Chutz L’tchum.
(if Issur Techumin is למעלה מעשרה )
The Shu”t Chasam Sofer (Chelek 6-98) explains
that there are two parts to .אליהו הנביא
His Guf (physical body) which is resting in Gan Eden
Hatachton, in this world. The other
part is his Neshama, which went up to heaven
and serves as a Malach.
As a Malach Habris (Not obligated in Mitzvos)
he can appear even on Shabbos
& Yom Tov. Therefore the Mohel can say
עמוד על ימיני even on Shabbos & Yom Tov.
In the near future when he will notify us of the
arrival of the Mashiach he will appear
physically as a human being . In such a state
he is Mechuyav in the Mitzvos and cannot
appear on Shabbos & Yom Tov.
The Aruch Hashulchan writes, The Minhag
at שפוך חמתך is to open the door
to remind us that it's ליל שימורם. In that זכות,
Mashiach will appear and will
be שופך חמתו on the( בבליים (גויים .
Poskim don't mention opening the
door for אליהו הנביא . According to the
Chasam Sofer אליהו הנביא won't appear
physically on Yom Tov and there is no
need to open the door for him*. The reason
to open the door is like the AruchHashulchan
writes" to remind us that it's ליל שימורם "
*For יציאת נשמה we do have a Minhag to open the window