Since Rosaline, the girl that Romeo is infatuated with at the opening of the play, is never an actual character in the play, there is no mention of what happens to her.
She is mentioned on the list of persons invited to the Capulet's feast, and this is the device that Shakespeare uses to contrive to have Romeo be in a position to actually meet and fall in love with Juliet. Rosaline is Capulet's niece, so not only does her existence as Romeo's first love provide the opportunity (Capulet's feast) for Romeo to meet Juliet, she also provides the precedent for him to fall in love with a girl from the House of his family's enemy.
The existence of Rosaline also provides the circumstance, in the opening Act of the play, for the audience to get a good picture of what kind of lover Romeo is. As he moons and mopes about the stage, they see how dramatic and poetical he is about love. These characteristics will present a nice foil to the more logical and practical minded lover that Juliet turns out to be. This is an ironic twist to the classic "boy/girl" lovers' character traits -- the young man who seems almost "girlie" in his approach to love and the young woman who is very logical, almost masculine in hers.
The Chorus enters before Act Two begins and says:
Now old affection [Rosaline] in his deathbed lie
And young affection [Juliet] gapes to be his heir
That fair for which loved groan'd for and would die,
With tender Juliet match'd is now not fair.
And from this moment the character of Rosaline is, in essence, "killed off." The play has no further use for her. And, as she is a fictitious character in a play, we, the audience, have no idea what "happens" to her.