I would suggest that one dominant moral from the tale of Pyramus and Thisbe would be that adults should not transfer their bitterness to children.
Pyramus and Thisbe are victims of their parents' animosity. The parents dislike one another with an intense passion. As a result, they forbid their children from pursuing their love. The sad tale of both young people are the products of this hatred. Both lovers kill themselves because they are forced to love one another in a covert manner. Had they been able to love one another with public transparency, their late night rendezvous and all else could have been avoided. In my mind, one of the morals of the myth is that children suffer when parents transfer hatred to them.
In terms of the lovers themselves, a moral could be to avoid rash action. Both Pyramus and Thisbe fail to reflect prior their actions. They respond in a rash and hasty manner. Pyramus kills himself thinking that the lioness has eaten his beloved. He does not pause to think things through and instead succumbs to quick action without contemplation. In much the same way, Thisbe responds with equal fervor. She sees the dead body of the man she loved and takes his sword to end her own life. She fails to reflect upon her actions, acquiescing to rapid decision making without introspection. Evident in the suicide of two young people, the sad ending of the narrative shows that little good results hasty action.
Pyramus and Thisbe tells the story of the two star-crossed lovers. Although family hatred separates them, Pyramus and Thisbe find a way to communicate through a hole in a wall. Soon, they plan to meet under a mulberry tree. Thisbe arrives first, but a lion chases her away. Her shawl is dropped under the tree, which the lion tears to shreds. Pyramus arrives, and seeing the blood-stained shawl, thinks the lion killed his lover. Pyramus commits suicide, and so does Thisbe when she sees her lover dying.
There are many morals of this myth. On one hand, one could argue that love (somewhat) prevails. Despite their parents' hatred towards another, Pyramus and Thisbe's love for each other causes them to go to certain measures to be with each other. On the other hand, another moral could be that every action has its consequences. Because they were blinded by each others' love, they went to extreme measures that resulted in death.