When I was brought up, I was told to face East when starting some new task or while doing prayers or chanting mantras.
My guess is that it might have come out of ancient sun (Surya) worship as sun rises in the east.
Is there any real significance for considering the Eastern direction to be auspicious?
It is because gods are believed to be present in the east direction of earth:
देवमनुष्या दिशो व्यभजन्त प्राचीं देवा दक्षिणा पितरः प्रतीचीं मनुष्या उदीचीं रुद्रा [Tait. Samh. - 6.1.1]
Meaning
Gods and men divided the directions. Gods took the east, pitaras (manes) took the south, men took the west and rudras took the north.
So while worshiping or praying to the gods, a person generally has to face to the east direction.
East is considered to be the cause of all kinds of knowledge, auspicious and prosperity.
The Mahabharata has a detailed significance of the East direction and thus, why must we face East. It gives all the major incidents associated with the East, and thus why it is considered the most auspicious.
Devarishi Narada narrates it as follows -
Chapter 108, Bhagavat-Yana Parva,
Udyog Parva, Mahabharata
O Galava, commanded I have been by God, who is the cause of all knowledge. I ask you, towards which quarter shall I first take you to see what lie there? The eastern, the southern, the western, or the northern, towards which, O best of regenerate persons, shall I go, O Galava? That quarter towards which Surya the illuminator of the universe first rises; where, at eve, the Sadhyas engage in their ascetic austerities; where that Intelligence, which pervades the whole universe first springs; where the two eyes of Dharma, as well as he himself, are stationed; where the clarified butter first poured in sacrifice subsequently flowed all around; that quarter,
O best of all regenerate persons, East is the gate of Day and Time. There the daughters of Daksha, in primeval times, gave birth to their children. There the sons of Kasyapa first multiplied.
That quarter is the source of all the prosperity of the gods, for it was there that Sakra was first anointed as the king of the celestials. It was there, O regenerate Rishi, that both Indra and the gods underwent their ascetic penances. It is for this, O Brahmana, that this quarter is called Purva (the first). And because in the earliest of times this quarter was overspread by the Suras, it is for this that it is called Purva. The gods, desirous of prosperity, performed all their religious ceremonies here.
It was here that the divine Creator of the universe first sang the Vedas. It was here that the Gayatri was first preached by Surya unto the reciters of that sacred hymn. It was here, O best of Brahmanas, that the Yajurvedas were delivered by Surya (unto Yajnavalkya). It was here that the Soma juice, sanctified by boons, was first drunk in sacrifices by Suras. It was here that the Homa-fires, (gratified by mantras), first drank articles of cognate origin.
It was here that Varuna first repaired to the nether regions, and attained to all his prosperity. It was here, O bull among the twice-born, that the birth, growth, and death of the ancient Vasishtha took place. Here first grew the hundred different branches of Om (different Vedas and Shrutis)!
It was here that the smoke-eating Munis are the smoke of sacrificial fires. It was in that region that myriads of boars and other animals were killed by Sakra and offered as sacrificial portions unto the gods. It is here that the thousand-rayed sun, arising, consumes, out of ire, all those that are wicked and ungrateful among men and the Asuras.
This (East direction) is the gate of the three worlds. This is the path of heaven and felicity. This quarter is called Purva (east).
And since East is the primordial source of all major incidents and scriptures within Hinduism, and is said to bestow prosperity and auspicious, thus East is the direction which is to be resorted to for any new task or daily start of the work-routine or day.
