Which gate will be closed on weekdays?
Rashi citing Midos 4:2: There were two Pishpeshim (small openings) to the gate of the Heichal, one to the north and one to the south. Above (44:2), it taught that no one will enter the southern one; it will be closed.
Malbim: Before teaching the Korbanos that will be offered during the Chinuch from Pesach until Sukos, it explains how the Nasi will come to the Beis ha'Mikdash to offer Korbanos. The gate of Chatzer ha'Penimis, i.e. the eastern gate of Ezras Yisrael, it will be closed the six days of the week, just like the eastern gate of Chatzer ha'Chitzonah. However, the outer gate will always be closed, and the inner gate will be open on Shabbos and Rosh Chodesh.
Michtav me'Eliyahu (5, p. 471): This is a gate in the heart. In Yemos ha'Mashi'ach, people will cling to Hashem and feel gates opening in the heart (Neshamah Yeserah) on Shabbos. Also on Rosh Chodesh, there will be a Chidush of influence of Kedushah ? this is the spiritual content of Rosh Chodesh.
What is "Chatzer ha'Penimis"?
Radak (1-2): It is Ezras Kohanim. The eastern gate of this Azarah is open to Ezras Yisrael. It will be closed during the six days of the week, for the nation and the Nasi do not come to the Beis ha'Mikdash then. It will be open only on Shabbos and Rosh Chodesh, and all the more so on Yamim Tovim. This will be a Chidush in the future.
Who will open the gate?
Yalkut Shimoni (383): It will open by itself, and everyone will know that it is Shabbos. Also on Rosh Chodesh, Yisrael will see it open, and know that the moon rose and upper beings are Mekadesh the month.
