In Othello, how does the willow tree song symbolize betrayal?

The willow tree song that Desdemona sings tells the story of a maid who dies because of love, of a broken heart.  As Desdemona sings it, the song merges into her own story, although she has no awareness that she will die soon, too, because of Othello's jealous love.  So, on the most obvious level, the song functions as foreshadowing. 

The willow tree is a popular symbol of sadness, often called the "weeping willow." As we tend to lean downward, in a slump, when we are sad or depressed, the tree, leaning downward, resembles a person in sadness.  Also, because of the way the branches look, the tree resembles a long-haired female leaning over, hiding her face in sadness.  There was a popular song in, I believe, the 1940's, called "Willow, Weep for Me,"  and I would guess that the willow has been used fairly frequently over the last several hundred years. For example, in the King James Version of the Bible, we have this:

By the rivers of Babylon, there we sat down, yea, we wept, when we remembered Zion.
We hanged our harps upon the willows in the midst thereof.

Other versions of the psalm reference poplars, so it is not at all clear what the original Hebrew was, but it is interesting to think that the willow was a symbol of sadness thousands of years ago. 

Desdemona's choice of this song shows us her sadness and gives us a sense of foreboding, not only because of the fate of the maid in the song, but also because the song relies heavily on the willow as a symbol of grief and sadness.  We can see the parallels between the song and Desdemona's life, and we know things are not going to turn out well.  

What does the song "Willow, willow" symbolize in Othello?

Your question refers to the song that Desdemona sings in Act IV scene 3 the night she is going to be murdered by her husband. What is important about this song is the way that Dedemona herself offers us the key to its symbolism. Note what she says about this song:

My mother had a maid called Barbary,
She was in love, and he she loved proved mad
And did forsake her. She had a song of "willow,"
An old thing 'twas, but it expressed her fortunes
And she died singing it.

What is important to realise is that this song possesses eerie echoes of her own situation, which is why it is so fitting to sing at this particular juncture. She too is in the position of worrying that the man she loves is losing is sanity and will leave her. In addition, let us remember that willows by water are traditionally a symbol of women who are left by their lovers. Consider how Ophelia in Hamlet drowns after Hamlet leaves her by falling out of a willow tree into water. The importance of the song therefore lies in the way that it allows Desdemona to express her own concerns and worries about her situation with Othello.