What is the relationship between Pyramus and Thisbe and A Midsummer Night´s Dream?

You have asked a really fascinating question. Lots of
people overlook the interesting choice of Shakespeare in including a tragedy as his play
within a play, and it is really important to consider the parallels between this play
and the action of the main play. Don´t overlook the importance of this choice by the way
the Mechanicals make a mockery of the tragedy!

For me,
Pyramus and Thisbe is included because it offers a kind of alternate ending of the
problem that we are presented with in Act I scene 1. Often, with Shakespeare´s plays,
the line dividing comedy from tragedy can be quite thin, and his plays have elements of
both. Act I scene 1 presents us with an opening that could be the start of a tragedy or
a comedy in my opinion. Hermia is given a very stark choice - submit to her father´s
will and marry Demetrius or face death. This parallels the situation between Pyramus and
Thisbe (also echoing Romeo and Juliet), as both couples have a love
that is not accepted by their parents or society at large. Both couples decide to run
away to the forest to elope and marry in secret away from society, but of course, with
Pyramus and Thisbe, this ends tragically, rather than the comic confusion with eventual
happy ending that the main play presents us with. Whilst the main characters mock and
poke fun at the Mechanicals for their over-the-top performance, the action they watch
bears a somewhat disturbing resemblance to their initial
problem.