The Babylonian story of Pyramus and Thisbe and the Shakespearean tragedy of Romeo and Juliet both mirror the theme of doomed lovers whose deadly fate begins with ancient grudges. While Pyramus and Thisbe's families are old enemies like Romeo and Juliet's and both pairs of lovers defy their families to be together, their doom comes to them in different ways. Pyramus, like Romeo, acts in haste when he finds the "bloody" cloak of Thisbe and like Romeo ends his life, though he uses a dagger and not poison. Thisbe, like Juliet, does not want to live without Pyramus when she finds him dead with her cloak and kills herself with the same dagger. Like Romeo and Juliet, Pyramus and Thisbe are immortalized at the story's end; however, instead of a statue in their memory erected by the parents, a mulberry tree's fruit grows from their blood and is forever stained red.
The original question had to be edited down. I think that the "doomed lovers" theme is evident in both narratives quite powerfully. The antagonisms forged in social division is evident in both, contributing to the "doomed" nature of each experience in love. Pyramus and Thisbe were next door to one another, but forbidden to see one another. This forbidding was imposed by a social order that was not of the young lover's making. For Romeo and Juliet, the Montague and Capulet fued is something that ripped Verona apart and was also not of their making. The lovers in both settings are doomed in part because of the social configuration that surrounds them. Another reason why they are doomed is because both sets of lovers believe that escape from this condition is possible. This helps to accentuate their tragic condition. Both Pyramus and Thisbe and Romeo and Juliet sincerely believe that their own actions can overcome the social conditions that are imposed upon them. Their respective failures to "beat the system" help to enhance their "doomed nature." Finally, the lack of convergence in both of their respective plans facilitates the idea that they are, in fact, doomed. Essentially, both of them escape, but lack of convergent timing and misunderstanding aids in how both are "doomed." Thisbe leaves and returns to find Pyramus dying. Juliet awakens to find Romeo dead. In both settings, the lovers are doomed because their timing, at its most critical moment, failed them. In these situations, one can see where the "doomed lovers" theme between both Pyramus and Thisbe and Romeo and Juliet is displayed.