When the Bais Hamikdash stood, on Yom Kipper the kohen gadol read from the torah facing the people and on Sukos the king read from the torah facing the people, why does our Bima face the Aron? If krias hatorah mimics matan torah (Shaarei Eefraim 376,10) shouldn't we face the people not the aron? What does the Bima represent that it faces the aron and not the people? Don't say the mizba'ach then it would need to face North (right of the aron).
Many Achronim including the Ma'adanei Yom Tov, Peri Megadim, Arukh Ha-shulchan, and Mishna Berura) write that one should not pray with one's back towards the Aron. Praying with your back towards the Aron is considered disrespectful as seen in the psukim : "their backs were to the House of God" (Yechezkel 8:16) and "they turned their backs towards Me" (Yirmiyahu 32:33). This prohibition takes precedence over the obligation to pray towards Jerusalem. So although i could not find any source that applies this rule to the reading of the torah, perhaps we don't read from the Torah with our backs to the Aron for the same reason.
Please review the following sources:
Now that you read the sources we still need to answer the following questions.
1) Why can you turn your back to the Aron during Lecha Dodi ?
2) It's common for Rabbis to say a drasha with their backs to the Aron. How ?
Lecha Dodi - user Chanoch (in the MY question linked above) notes that Rav Ovadiah OB"M says in Yalkut Yosef that it is permitted.
Drashot - A) Many Rabbis do indeed say their Drashot off to the side of the Aron Kodesh B) I have seen many Rabbis who when they have to say a Drasha with their backs turned to the Aron first kiss the Parochet. This may be to show the tzibor of listeners that no disrespect is meant to the Aron. C) Perhaps they are relying on the lenient position mentioned above.
It is obvious that these are exceptions to the rule, otherwise why would it mention by a the Kohain Gadol and King that they face the crowd.