How Can He Appear ?

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