Under what circumstances is it permissible or forbidden to sit or stand in the synagogue with one's back facing the Aron (ark)?
The Gemara in Masechet Yoma describes the procedure followed in the Beit Ha'mikdash when conducting the daily "Payis," the lottery that would determine which Kohanim would perform the various rituals each day. As the Gemara describes, the "Payis" was conducted in the Lishkat Ha'gazit, a chamber situated off the Azara, the courtyard of the Temple. The Rashbam (Rabbi Shemuel Ben Meir, grandson of Rashi, France, 1085-1174) offers two reasons why the "Payis" was held away from the sacred area of the Azara. Firstly, the administrator of the Kohanim who conducted the lottery would begin by removing the hat of one of the Kohanim, and it would be disrespectful for that Kohen to remain bareheaded in the Azara. Secondly, the Rashbam explains, the Kohanim stood in a circle during the lottery, and therefore if the lottery would take place in the Azara, invariably some Kohanim would be standing with their back to the Sanctuary of the Beit Ha'mikdash. The Kohanim avoided this problem by conducting the lottery inside the Lishkat Ha'gazit.
On the basis of the Rashbam's discussion, the Leket Yosher (biography and rulings of Rabbi Yisrael Isserlin, Germany-Austria, 1390-1460 – vol. 1, p. 31) rules that it is forbidden to stand or sit with one's back to the Aron in the synagogue. Just as the Kohanim had to avoid standing with their backs to the Sanctuary in the Temple, so does Halacha require that one avoid standing or sitting with his back to the ark in the synagogue. Therefore, as the Mishna Berura (commentary to the Shulchan Aruch by the "Chafetz Chayim," Rabbi Yisrael Kagan, Lithuania, 1835-1833) rules (150:14), seats in a synagogue should not be positioned directly in front of the ark with their back to the ark. In some synagogues the Rabbi sits facing the congregation; this is permissible provided that his chair is situated to the side of the Aron, and not directly in front of it.
During Birkat Kohanim, the Kohanim are permitted to stand with their backs to the Aron, facing the congregation, in the interest of Kevod Tzibur – showing respect to the congregation.
Furthermore, when the Torah is being brought from the ark to the Bima (table used for Torah reading), one may turn his back to the Aron for the purpose of following the Torah, in order to give honor to the Torah.
The Peri Megadim (Rabbi Yosef Te'omim, Poland-Germany, 1727-1792) raises the question as to whether people sitting in between the Bima and Aron should face the Aron during the Torah reading, given that their backs will thus be turned to the Torah. He writes that they should nevertheless face the ark, since the Bima itself constitutes an interruption between them and the Torah on the Bima, such that we do not consider them as turning their backs to the Torah.
Summary: One must not sit or stand in front of the Aron in the synagogue with one's back facing the Aron, except for Kohanim while they recite Birkat Kohanim, and as one follows the Torah from the ark to the Bima. During Torah reading, when the Torah is situated on the Bima, people sitting in between the ark and the Bima should nevertheless face the ark, with their backs to the Bima.
