Who is the “old desire” referred to in the first line?
The "old desire" does not exactly refer to a person but to Romeo's feelings for a person. When the play opens he is suffering from the pangs of love for a girl named Rosaline. But Rosaline has no interest in Romeo or in any other young man. He tells Benvolio:
She hath Dian's wit,
And, in strong proof of chastity well armed,
From Love's weak childish bow she lives unharmed.
She will not stay the siege of loving terms,
Nor bide th' encounter of assailing eyes,
Nor ope her lap to saint-seducing gold.
O she's rich in beauty, only poor
That when she dies, with beauty dies her store.
His passion for this seemingly cold young woman is the "old desire" referred to in the Prologue to Act 2. The Chorus begins with these lines:
Now old desire doth in his deathbed lie,
And young affection gapes to be his heir.
Obviously "old desire" could not be referring to a person, especially to a young girl. We can hardly imagine Rosaline lying on her deathbed. What has happened since the events dramatized in Act 1 is that Romeo has met Juliet at the Capulet's party and has fallen in love at first sight. Juliet is hardly as cold as Rosaline, though Juliet is only thirteen. She is not the "new desire," but she has awakened a new desire in Romeo, and he has completely forgotten about Rosaline. Her function in the play may have been to serve as a contrast to Juliet, who is more than willing to give Romeo that total love he is seeking.