I think when most people hear "tragedy" as a literary genre, the first thing that comes to mind is death, which is--ultimately--one similarity between these two plays, though probably not the greatest. While there are definitely distinct differences between Greek tragedy and Shakespearean tragedy it should be noted that Romantic tragedies (Shakespeare at the forefront) drew from the traditional tragic elements that started with the Greeks but then twisted these elements to create something new.
The similarities, therefore, come down to basics. Both "Oedipus Rex" and "Romeo and Juliet" share the element that everyone knows from the very outset that the play will end tragically (in this way, they both lack surprise endings). The irony of "Oedipus Rex" was based on the fact that the audience knows all along that Oedipus is guilty and headed toward ruin. In the same way, "Romeo and Juliet" opens with a prologue that foreshadows the deaths of the main characters.
Also, though the concept of the "tragic hero" started in Greek tragedy, it could be argued that the characters in "Romeo and Juliet" are very similar to King Oedipus in heroic qualities and fatal flaws. Both Romeo and Juliet possess many of the traditional elements that make up a tragic hero (including nobility by birth, aiming at propriety, true to life, possessing a tragic flaw that causes their downfalls, and evoking pity in the reader). Somewhat against the notion of the "fatal flaw" and "fall," others have argued that all three principle characters of these two plays were helpless victims of fate. The very idea of "fate" as a factor is common to both plays.
Other elements that "Oedipus" and "Romeo and Juliet" share, as dramas, are the use of a chorus (throughout Oedipus and in the Prologue of Romeo and Juliet) who speaks in "odes" or "verse" and the use of light/darkness symbolism (including blindness). Finally, it should be noted, both plays certainly evoke pity (among other strong emotions) from the audience or reader, which was one of the original and primary purposes for tragedy.