What is the biblical basis for the Catholic and Eastern doctrines of the Assumption of Mary? Catholic doctrine asserts that Mary the Mother of Jesus was not only without sin and ever-virgin, but that she also ascended into Heaven. Some Eastern traditions hold that she even died, then was raised on the third day. 

The Catechism of the Catholic Church States, in paragraph 974:


  The Most Blessed Virgin Mary, when the course of her earthly life was completed, was taken up body and soul into the glory of heaven, where she already shares in the glory of her Son's Resurrection, anticipating the resurrection of all members of His Body.


From Wikipedia:


  Eastern Christians believe that Mary died a natural death, that her soul was received by Christ upon death, and that her body was resurrected on the third day after her death and that she was taken up into heaven bodily in anticipation of the general resurrection. Her tomb was found empty on the third day. "...Orthodox tradition is clear and unwavering in regard to the central point [of the Dormition]: the Holy Virgin underwent, as did her Son, a physical death, but her body – like His – was afterwards raised from the dead and she was taken up into heaven, in her body as well as in her soul. She has passed beyond death and judgement, and lives wholly in the Age to Come. 


What is the biblical basis for this doctrine?
There is no obvious basis for the Assumption of Mary in the Bible. Mary is mentioned a few times outside the birth narratives of Luke and Matthew:

Is he not the carpenter, the son of Mary... (Mark 6:3)
Is he not the carpenter's son? Is not his mother named Mary ... (Matthew 13:55)
Standing by the cross of Jesus were his mother ... (John 19:25)
On the third day there was a wedding in Cana in Galilee, and the mother of Jesus was there. (John 2:1)
His mother and his brothers arrived. (Mark 3:31)
While he was still speaking to the crowds, his mother and his brothers appeared (Matthew 12:46)
Then his mother and his brothers came to him (Luke 8:19)
All these devoted themselves with one accord to prayer, together with some women, and Mary the mother of Jesus (Acts 1:14)

Nothing is said about Mary's death specifically. However, it appears from this link given in another answer that the Church traditionally has interpreted the vision of the woman in Revelation 12 as referring to Mary taken up into heaven:

A great sign appeared in the sky, a woman clothed with the sun, with the moon under her feet, and on her head a crown of twelve stars. ... The woman herself fled into the desert where she had a place prepared by God ...

(Revelation 12:1,6a)
This may be as close as we get to a biblical basis for the belief in the Assumption.
Since you specifically do not ask about the history, I am not including another review of that.