Is there any biblical evidence that suggests Jesus read or wrote anything, and was he literate?
Jesus read Isaiah 61 in a synagogue:
Luke 4:16-22 (ESV)
16 And he came to Nazareth, where he had been brought up. And as was his custom, he went to the synagogue on the Sabbath day, and he stood up to read. 17 And the scroll of the prophet Isaiah was given to him. He unrolled the scroll and found the place where it was written, ... 20 And he rolled up the scroll and gave it back to the attendant and sat down. And the eyes of all in the synagogue were fixed on him. 21 And he began to say to them, “Today this Scripture has been fulfilled in your hearing.” 22 And all spoke well of him and marveled at the gracious words that were coming from his mouth. And they said, “Is not this Joseph's son?”
Jesus wrote on the ground:
John 8:6-8 (ESV)
6 This they said to test him, that they might have some charge to bring against him. Jesus bent down and wrote with his finger on the ground. 7 And as they continued to ask him, he stood up and said to them, “Let him who is without sin among you be the first to throw a stone at her.” 8 And once more he bent down and wrote on the ground.
In addition to the direct implications of these, we have less direct but still strong evidence in events such as his delivery of the Sermon on the Mount. (Matthew 5-7). Also, if you are a Christian, you believe he co-authored the entire Bible (Hebrews 1:1-2).
Times change, and the exact measure that defines "literacy" changes with them. The concept of an "essay" would have been unknown in Jesus' day, but it is quite clear from the record of we have of his interaction with other learned men that he was treated as if he was educated. As we learn from the verses quoted in parap's answer he was clearly able to read the Tanakh scrolls. These two factors are enough for us to label him as being literate. Living and ministering in a primarily oral culture, it is not surprising that Jesus chose not to sit and write books but rather preached with parables to the public and by reasoning from the Scriptures with those educated in them.
Matthew 13:54 (ESV)
54 and coming to his hometown he taught them in their synagogue, so that they were astonished, and said, “Where did this man get this wisdom and these mighty works?
The verse here says that Jesus astonish them learned men of his day with his wisdom and he taught them. This is not the work of an unlearned man. Teaching from, reasoning from and accurately quoting the OT Scriptures (dozens of times) are clear signs of solid education and literacy skills.
John 8:6 would appear to show that Jesus could write:
John 8:6: This they said, tempting him, that they might have to accuse him. But Jesus stooped down, and with his finger wrote on the ground, as though he heard them not.
However, this passage, known as the Pericope Adulterae is regarded as inauthentic by the great majority of biblical scholars, being absent from most early manuscripts, as well as being found in different locations when it does occur. It provides no reliable biblical evidence that Jesus could write, and the presence of this verse could merely indicate a desire among later scholars to prove that Jesus could indeed read and write.
In The Birth of Christianity, pages 234-5, John Dominic Crossan cites evidence from rabbinic sources that support a conclusion that the literacy rate in the Jewish homeland was less than 3 per cent. He says this literacy rate, though low by modern standards, was not low at all taking into account the needs of a traditional society in the past.
Crossan points out Jesus was a peasant from a peasant village. Therefore, for Crossan, Jesus was illiterate until the opposite is proven. It is not proven but simply presumed by Luke, when he has Jesus read from the prophet Isaiah in the synagogue at Nazareth in 4:16-20 (Remembering that Luke, if he was the author of the gospel that now bears his name, appears never to have met Jesus). Jesus did not write, and in the opinion of Crossan, could not write.