What is the biblical basis for the Catholic and Eastern doctrines of the Assumption of Mary?

There is no Biblical basis to the Assumption of Mary.

the Bible seems to be silent about it, and even the early fathers of the Church say nothing about it.

It was an apocryphal belief established in the 4th century, and later added to Catholic canon in the 1950's.

Although the Assumption (Latin: assumptio, "a taking") was only relatively recently defined as infallible dogma by the Catholic Church, and in spite of a statement by Saint Epiphanius of Salamis in AD 377 that no one knew whether Mary had died or not,[7] apocryphal accounts of the assumption of Mary into heaven have circulated since at least the 4th century.

There are no remaining records of the belief from before that time, and it appears to have originated from the pagan beliefs of the areas Christianity had spread into.

The origin of this idolatry had its root in ancient mythology. Astarte of the Assyrians, Ashtoreth of the Sidonians and Bowaney of the Hindoos held the place that Mary occupies in the church of Rome. Greece had her Venus and Rome her Juno. The Diana of the Ephesians was a female, from whose body in every part there seemed to be issuing all the various animals of creation, symbolizing the conception and creation of all things.

Are the Marian Doctrines explicitly mentioned in the Bible? The answer is "no". It is worth remembering, however, that every Christian believes something not explicitly mentioned in the Bible. We are thus brought to the interrelated questions of authority and interpretation. In other words, we must ask: Who has the ultimate authority to interpret Revelation?

However, before I say anything about these issues, I would like to address the charge of idolatry. The Marian Doctrines, though not strictly Biblical, are emphatically not pillars of idolatry. If the Catholic Church taught that Mary is equal to God, it would be guilty of idolatry. While the Catholic Church does teach that Mary is the new Eve, and while Mary does enjoy a privileged position within Catholic theology, the Church has never taught that Mary is God.

Concerning the question of authority, it is worth noting at the outset that the existence of the Church far predates the existence of a canonical Bible. Nevertheless, the Bible is crucial to Christian life as we know it -- and anything which contradicts the Bible should be thrown away. Still, we should not commit the heresy of believing that the Word can be reduced to the Bible. After all, the Word of God is Jesus Christ Himself. Jesus is Revelation made flesh. Having said that, who has the authority to interpret Scripture? You? Me? Your Pastor? Who gets to declare what is really God's saving truth?

Some Christians feel there is no need for interpretation. Aren't things pretty clear? Let's consider an example:

If I washed your feet -- I who am Teacher and Lord -- then you must wash each other's feet. What I just did was to give you an example: as I have done, so you must do. (John 13:14)

This seems pretty clear! Where is the need for interpretation? But how many Christians wash the feet of others? Some Christians claim that Jesus was acting symbolically: Washing feet represents service to others. This is an alright view -- but it is an interpretation. Where in the Gospels does it explicitly say that Jesus was acting symbolically? Some would argue it's implied. Fine. But again, we come against the issue of interpreting Scripture, which always involves ideas not explicitly mentioned in the Bible. If you interpret this passage symbolically, you make use of an authority outside the Bible. Maybe it's your own interpretation. Maybe it's your Pastor's. In any case, we begin to see that referring to the Bible in and of itself as the ultimate authority is questionable.

Catholics know this. Catholics admit this. They see the ultimate authority as residing with the Church. After all, the Church not only interprets the Bible -- it gave us the Bible.

And this does tie in with what we know about Jesus' ministry. After all, one of His first acts (with respect to His ministry) was to gather the 12, representing the 12 tribes of Israel. Jesus never promoted individualistic spirituality; but sought to bring the People of God together to properly hear God's Word.