What is the Biblical evidence against the perpetual virginity of Mary? I understand that the Roman Catholic Church teaches that Mary is the eternal virgin in that she never had sexual relations with Joseph (or any other man)--either before or after the miraculous conception of Jesus.

What Biblical evidence is there that contradicts this doctrine?
One such passage is Matthew 13:55, where Jesus is identified thus:


  “Isn’t this the carpenter’s son? Isn’t his mother’s name Mary, and
  aren’t his brothers James, Joseph, Simon and Judas?"


Some would quibble over the definition and translation of the word "brother" here, taking it to mean that the brothers listed weren't the offspring of Mary but spiritual brothers (with no genetic link) or perhaps half-brothers (which would require Joseph to have had another wife - there's no evidence of this) or other relations. But much more definitive is Matthew 27:56, where Mary is described as the mother of those first two brothers:


  Among them were Mary Magdalene, Mary the mother of James and
  Joseph, and the mother of Zebedee’s sons.


Interestingly Mary here isn't identified as the mother of Jesus as well as James and Joseph, so one possible rebuttal to this verse is that Mary the mother of James and Joseph is a different Mary from Mary the mother of Jesus.

There are similar passages elsewhere in the New Testament that refer to Jesus' mother and his brothers, the context and normal processes of hermeneutics would lead to the conclusion that Mary (Jesus' mother) did indeed have these other offspring - but, to defend the Catholics a little bit, none of those passages state unambiguously that his brothers were born of his mother.

One further argument that Mary had further children after Jesus is that Jesus is referred in the Bible as Mary and Joseph's "firstborn", and never as their only child.

Matthew 1:25 (AMP):


  But he had no union with her as her husband until she had borne her
  firstborn Son; and he called His name Jesus.


But to be pedantic, an only child is also a firstborn, so this too could be argued against.

In summary, there is very strong Biblical evidence that Mary went on to have other children and therefore did not remain a virgin after the birth of Jesus; certainly that is the natural conclusion to be drawn from what is written in the Bible. However there are ways  of challenging the wording and interpretation of each of the verses above in order to form an argument to the contrary.

Reference/further reading: "Did Jesus Have Brothers and Sisters?" by Tony Warren