Love (and its opposite, hate) is one of the primary themes in Romeo and Juliet by William Shakespeare. Your prompt statement, "love does not last," represents one aspect of love in this play.
In Act I, we observe several instances of this statement. First of all, we meet a depressed, languishing Romeo, pining away in the darkness for the love of his life, Rosaline. That night, he goes to the Capulet ball and meets Juliet. His love for Rosaline is suddenly and completely gone. That love did not last.
We also learn from Juliet's father that the love he and his wife had (they were married when Lady Capulet was younger than Juliet) is now something, well, less than love. Capulet says this about marriages made too early:
And too soon marred are those so early made.
His implication is that the "love" is gone.
In Act III scene ii, Juliet learns that Romeo, now her husband, has killed her cousin Tybalt. She says:
A (damned) saint, an honorable villain!
O nature, what hadst thou to do in hell
When thou didst bower the spirit of a fiend
In mortal paradise of such sweet flesh?
For a few moments, anyway, Juliet wavers between love and hate for Romeo.
In a different sense, love does not last because of death. The Nurse has lost her husband, Romeo loses a friend (Mercutio) and Juliet loses her cousin (Tybalt).
A better question might be to think about the changing face of love in this story, but certainly the idea that "love does not last" is found in the first three acts--and it is an even more compelling idea for the last two acts of the play.