This question is presumably in reaction to Nicene Christianity, which holds that there is a Trinity of Persons consisting of God the Father, God the Son, and God the Holy Spirit, each of which is eternal, the Son being eternally begotten from the Father, or born from eternity, and the Holy Spirit proceeding eternally from the Father (and the Son). It is therefore best answered in response to and in contrast with that doctrine.

There is no mention in the Bible of a Son born from eternity

The first and primary biblical evidence against a pre-incarnate Jesus, or Son of God, is the simple lack of any statement to that effect anywhere in the Bible.

Yes, Jesus did make statements such as:

Very truly, I tell you, before Abraham was, I am. (John 8:58)

and:

So now, Father, glorify me in your own presence with the glory that I had in your presence before the world existed. (John 17:5)

Nicene Christianity interprets such passages to mean that there is a Son born from eternity. However, in no place does the Bible ever state that there was a Son from eternity, or a Son born from eternity, or that Jesus as Jesus pre-existed his conception in the womb of Mary.

That concept and language probably developed in the fourth century, as reflected in the progression of Creeds generally accepted in Nicene Christianity.

It is not stated in the Apostles' Creed, commonly thought to be the earliest of the existing Creeds in its origins. That Creed speaks only of Jesus as being conceived by the Holy Spirit and born of the Virgin Mary, with no mention of his having any pre-existence:

I believe in Jesus Christ, God's only Son, our Lord,
who was conceived by the Holy Spirit,
born of the Virgin Mary,
suffered under Pontius Pilate,
was crucified, died, and was buried;
he descended to the dead.
On the third day he rose again;
he ascended into heaven,
he is seated at the right hand of the Father,
and he will come to judge the living and the dead.

(Source: Wikipedia -> Apostles' Creed -> English translations)

It is also not stated in the original 325 AD version of the Nicene Creed. However, the revised 381 AD version of the Nicene Creeds adds the words "before all worlds":

And in one Lord Jesus Christ, the only-begotten Son of God, begotten of the Father before all worlds (æons), Light of Light, very God of very God, begotten, not made, being of one substance with the Father;

(Source: Wikipedia -> Nicene Creed -> Comparison between Creed of 325 and Creed of 381. Italics indicated wording added to the 381 version that was not in the 325 version)

The Greek phrase traditionally translated "before all worlds" is πρὸ πάντων τῶν αἰώνων, which in more contemporary English, means "before all ages."

In the Athanasian Creed, which more precisely defines the Trinity of Persons, and is accepted in most of Nicene Christianity, the eternity of the Son is expressed in these terms:

For there is one Person of the Father; another of the Son; and another of the Holy Ghost. But the Godhead of the Father, of the Son, and of the Holy Ghost, is all one; the Glory equal, the Majesty coeternal. Such as the Father is; such is the Son; and such is the Holy Ghost. The Father uncreated; the Son uncreated; and the Holy Ghost uncreated. The Father unlimited; the Son unlimited; and the Holy Ghost unlimited. The Father eternal; the Son eternal; and the Holy Ghost eternal. . . . And in this Trinity none is before, or after another; none is greater, or less than another. But the whole three Persons are coeternal, and coequal. (italics added)

(Source: Wikipedia -> Nicene Creed -> Content)

So as stated above, the first piece of biblical evidence against a pre-incarnate Jesus is simply that the language commonly used to express that concept does not occur anywhere in the text of the Bible, but only in creedal statements composed several centuries after the last books of the Bible were written.

There is no mention of any Son of God in the Old Testament

If Jesus, the Son of God, existed before his conception in the womb of Mary, we would expect to find some mention of him in the Old Testament.

However, aside from some mentions of "sons of God" and "children of God" referring to angelic or human beings, there is no mention of any "Son of God" in the Old Testament. That term first comes into use in the Gospels, and it always refers to Jesus Christ, who was, of course, alive on earth at the time of the events being narrated.

Instead, the Old Testament consistently speaks of one God, who is the Lord, the Savior, the Redeemer, and has all of the other attributes later attributed to Jesus Christ in the New Testament. Here are just a few of many, many examples that could be given:

Let the words of my mouth and the meditation of my heart
    be acceptable to you,
    O Lord, my rock and my redeemer.
                                           (Psalm 19:14)

O Israel, hope in the Lord!
    For with the Lord there is steadfast love,
    and with him is great power to redeem.
It is he who will redeem Israel
    from all its iniquities.
                               (Psalm 130:7-8)

Thus says the Lord, the King of Israel,
    and his Redeemer, the Lord of hosts:
I am the first and I am the last;
    besides me there is no god.
                           (Isaiah 44:6)

Declare and present your case;
    let them take counsel together!
Who told this long ago?
    Who declared it of old?
Was it not I, the Lord?
    There is no other god besides me,
a righteous God and a Savior;
    there is no one besides me.
Turn to me and be saved,
    all the ends of the earth!
    For I am God, and there is no other.
                       (Isaiah 45:21-22)

Yet I have been the Lord your God
    ever since the land of Egypt;
you know no God but me,
    and besides me there is no savior.
                          (Hosea 13:4)

Though Nicene Christians may read a Trinity of Persons into these statements, the statements themselves are not trinitarian, but speak of one God who is the Lord, Redeemer, and Savior, with no mention of any Son or Holy Spirit as distinct Persons of God.

And of course Judaism, which accepts only the Hebrew Bible (which Christians call the Old Testament) as Scripture, has never believed in or accepted any concept of a Trinity in God because there are no statements of any such Trinity in the Hebrew Bible.

So a second piece of biblical evidence against a pre-incarnate Jesus is that there is no mention of any such Person or Being in the entirety of the Old Testament, which covers the time period in which Jesus, the Son, would have pre-existed. Instead, God is presented in the Old Testament as being absolutely and eternally one, and as encompassing all of the attributes that are attributed by Nicene Christians severally to the various Persons of the Trinity.

Jesus is presented in the Bible as being conceived and born in time

As stated above, the Bible never speaks of any pre-incarnate Jesus, or any Son born from eternity. But it does speak of Jesus being born at a particular time and place:

Now the birth of Jesus the Messiah took place in this way. When his mother Mary had been engaged to Joseph, but before they lived together, she was found to be with child from the Holy Spirit. (Matthew 1:18)

In the sixth month the angel Gabriel was sent by God to a town in Galilee called Nazareth, to a virgin engaged to a man whose name was Joseph, of the house of David. The virgin’s name was Mary. And he came to her and said, “Greetings, favored one! The Lord is with you.” But she was much perplexed by his words and pondered what sort of greeting this might be. The angel said to her, “Do not be afraid, Mary, for you have found favor with God. And now, you will conceive in your womb and bear a son, and you will name him Jesus. He will be great, and will be called the Son of the Most High, and the Lord God will give to him the throne of his ancestor David. He will reign over the house of Jacob forever, and of his kingdom there will be no end.” Mary said to the angel, “How can this be, since I am a virgin?” The angel said to her, “The Holy Spirit will come upon you, and the power of the Most High will overshadow you; therefore the child to be born will be holy; he will be called Son of God.” (Luke 1:26-35)

Further, although the Old Testament sometimes poetically speaks in the present tense of a Son being born to us (Hebrew language structure and poetry commonly has a sense of timelessness about it), the coming of the Messiah, or of God to earth, is presented in the Old Testament as a future event. For example:

Therefore the Lord himself will give you a sign: The virgin will conceive and give birth to a son, and will call him Immanuel. (Isaiah 7:14)

This prophecy is explicitly applied to Jesus in the New Testament:

All this took place to fulfill what the Lord had said through the prophet: “The virgin will conceive and give birth to a son, and they will call him Immanuel” (which means “God with us”). (Matthew 1:22-23)

Another Old Testament prophecy that Christian take as applying to Jesus Christ:

For to us a child is born,
    to us a son is given,
    and the government will be on his shoulders.
And he will be called
    Wonderful Counselor, Mighty God,
    Everlasting Father, Prince of Peace.
                                    (Isaiah 9:6)

Although this is set in the aforementioned present tense common in Hebrew poetry, it is referring to a future event, as explicitly stated in the introduction to the prophecy:

Nevertheless, there will be no more gloom for those who were in distress. In the past he humbled the land of Zebulun and the land of Naphtali, but in the future he will honor Galilee of the nations, by the Way of the Sea, beyond the Jordan— (Isaiah 9:1, italics added)

And just one more for now (dozens more could be quoted):

I will set his hand on the sea
    and his right hand on the rivers.
He shall cry to me, “You are my Father,
    my God, and the Rock of my salvation!”
I will make him the firstborn,
    the highest of the kings of the earth.
                          (Psalm 89:25-27)

Here the firstborn of God, and the highest of the kings of the earth—whom Christians, of course, believe to be Jesus Christ, the Son of God—is presented as coming in the future, not as existing in the present.

So a third piece of biblical evidence against a pre-incarnate Jesus is that in the New Testament Jesus is presented as being born at a particular time and place, while in the Old Testament the coming of the Messiah, or of the Lord, is presented as a future event. There is no mention anywhere in the Bible of the Messiah, or Jesus, existing before his conception in the womb of Mary.

The Bible does not state or require a Son born from eternity

The above three points show that the Bible itself makes no mention of any Son born from eternity, or of Jesus existing as Jesus prior to his conception in the womb of Mary. Rather, the Old Testament presents the Messiah and the coming of God as a future event, and the New Testament presents Jesus as being conceived and born at a particular time and place.

The language of Bible itself, then, does not speak of or require a Son born from eternity, or a pre-existing Jesus, the Messiah.

Nicene Christians commonly believe that only their concept of a Trinity of coeternal Persons of God, of which the second, the Son, is "eternally born" of the Father, satisfies the various statements about Jesus, the Son of God and the Messiah, that occur in the New Testament. However there are other, non-Nicene Christian theologies that do not hold to an eternally-begotten Son, and that read and interpret differently the Bible passages in which Jesus speaks of having glory with the Father before the world existed (John 17:5), of the Word being with God and being God in the beginning and then becoming flesh (John 1:1-4, 14), and so on.

Presenting such a theology is beyond the scope of the question. However, such a theology would include a belief that although the Word (Greek λόγος, logos) was eternally with God and was God as stated in John 1:1, it was not Jesus, or the Messiah, or the Son of God, until it "became flesh and lived among us" as stated in John 1:14—that it was in the act of the Incarnation, of "becoming flesh," that the eternal Word became Jesus, the Messiah, and the Son of God.

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Evidence for denying a pre-incarnate son
There are a number of verses, but for now, I'll focus on one the majority of Christians look to, and that's John 1:1. The way a trinitarian might understand this verse is that the word is God's son. A unitarian understands the word as the word that is personified by John, similarly to the way wisdom is personified in Proverbs 8:22-31. Here are few verses regarding God's word personified.

• God spoke to create the Universe. (Genesis 1; Psalm 33:6, 9)
• God's word is sent out to accomplish that which God wills. (Isa. 55:10, 11)

Jesus' pre-existence and biblical unitarian positions
Jesus the man and messiah is made a reality when the word became flesh. (John 1:14) When it comes to biblical unitarianism, there are different views. One common belief they all hold is that Jesus isn't the god known as Yahweh. This belief in a single, unipersonal god is exclusive to the Father.