What is the biblical basis in the Old Testament for the belief that the Messiah must be God?

In the Old Testament there are two general strains of prophecy commonly read by Christians as being about the Messiah who was to come:

1. Prophecies of a king in the line of David
2. Prophecies of God himself coming to his people

Judaism generally accepts the first class of prophecy as being about the prophesied Messiah, but not the second. And a strict reading of the Old Testament for passages mentioning an "anointed one" (Hebrew: מָשִׁיחַ mashiyach) and a king in the line of David generally does support this interpretation—though there are some passages that can be read as fitting into both classes of prophecy at once.

Christians commonly accept the second class of prophecies as also being about the Messiah to come, even when they are not specifically linked to a coming anointed king in the line of David.

Here are some examples prophecies that fall into this second class, or span both classes:

In the Psalms:

Lift up your heads, you gates;
    be lifted up, you ancient doors,
    that the King of glory may come in.
Who is this King of glory?
    The Lord strong and mighty,
    the Lord mighty in battle.
Lift up your heads, you gates;
    lift them up, you ancient doors,
    that the King of glory may come in.
Who is he, this King of glory?
    The Lord Almighty—
    he is the King of glory.
                        (Psalm 24:7–10)

In Isaiah:

The people who walked in darkness
    have seen a great light;
those who lived in a land of deep darkness—
    on them light has shined.
You have multiplied the nation,
    you have increased its joy;
they rejoice before you
    as with joy at the harvest,
    as people exult when dividing plunder.
For the yoke of their burden,
    and the bar across their shoulders,
    the rod of their oppressor,
    you have broken as on the day of Midian.
For all the boots of the tramping warriors
    and all the garments rolled in blood
    shall be burned as fuel for the fire.
For a child has been born for us,
    a son given to us;
authority rests upon his shoulders;
    and he is named
Wonderful Counselor, Mighty God,
    Everlasting Father, Prince of Peace.
His authority shall grow continually,
    and there shall be endless peace
for the throne of David and his kingdom.
    He will establish and uphold it
with justice and with righteousness
    from this time onward and forevermore.
The zeal of the Lord of hosts will do this.
                              (Isaiah 9:2–7)

And another in Isaiah:

The Lord saw it, and it displeased him
    that there was no justice.
He saw that there was no one,
    and was appalled that there was no one to intervene;
so his own arm brought him victory,
    and his righteousness upheld him.
He put on righteousness like a breastplate,
    and a helmet of salvation on his head;
he put on garments of vengeance for clothing,
    and wrapped himself in fury as in a mantle.
According to their deeds, so will he repay;
    wrath to his adversaries, requital to his enemies;
    to the coastlands he will render requital.
So those in the west shall fear the name of the Lord,
    and those in the east, his glory;
for he will come like a pent-up stream
    that the wind of the Lord drives on.
And he will come to Zion as Redeemer,
    to those in Jacob who turn from transgression, says the Lord.
                                                (Isaiah 59:15–20)

And yet another in Isaiah:

"Who is this that comes from Edom,
    from Bozrah in garments stained crimson?
Who is this so splendidly robed,
    marching in his great might?"
"It is I, announcing vindication,
    mighty to save."
"Why are your robes red,
    and your garments like theirs who tread the wine press?"
"I have trodden the wine press alone,
    and from the peoples no one was with me;
I trod them in my anger
    and trampled them in my wrath;
their juice spattered on my garments,
    and stained all my robes.
For the day of vengeance was in my heart,
    and the year for my redeeming work had come.
I looked, but there was no helper;
    I stared, but there was no one to sustain me;
so my own arm brought me victory,
    and my wrath sustained me.
I trampled down peoples in my anger,
    I crushed them in my wrath,
    and I poured out their lifeblood on the earth."

I will recount the gracious deeds of the Lord,
    the praiseworthy acts of the Lord,
because of all that the Lord has done for us,
    and the great favor to the house of Israel
that he has shown them according to his mercy,
    according to the abundance of his steadfast love.
For he said, "Surely they are my people,
    children who will not deal falsely";
and he became their savior
    in all their distress.
It was no messenger or angel
    but his presence that saved them;
in his love and in his pity he redeemed them;
    he lifted them up and carried them all the days of old.
                                            (Isaiah 63:1–9)

In Jeremiah:

"The days are coming," declares the Lord,
    "when I will raise up for David a righteous Branch,
a King who will reign wisely
    and do what is just and right in the land.
In his days Judah will be saved
    and Israel will live in safety.
This is the name by which he will be called:
    The Lord Our Righteous Savior."
                                      (Jeremiah 23:5–6)

In Malachi:

See, I am sending my messenger to prepare the way before me, and the Lord whom you seek will suddenly come to his temple. (Malachi 3:1)

These are just a few of many passages in the Old Testament that are read by Christians as applying to the prophesied Messiah, and as saying that the Messiah will not be a mere human being, but will be God himself.