Many, many different sins are committed by many different sinners throughout Romeo and Juliet. Below is a discussion of a couple.
One of the most important sins is committed by the sinners of Lords Capulet and Montague. The Prologue makes it evident that the feud between the Montagues and Capulets is an ancient feud, but also that it had died down but has now been rekindled by the present generation, as we see in the line, "From ancient grudge break to new mutiny" (Prologue, 3). As Prince Escalus points out, this new outbreak has already caused "[t]hree civil brawls" due to nothing more than an "airy word" or "trivial" comment (eNotes, I.i.85). The brawls have already led to a lot of blood loss and, during the coarse of the play, leads to several deaths, including Tybalt's and Mercutio's, as well as Romeo's and Juliet's. It is their pride and their violent, uncontrolled tempers that have caused the brawls and the destruction; pride and uncontrolled emotions are sins by themselves leading to the further sin of indirect murder.
Friar Laurence, unfortunately, is another sinner in the play; although, unlike Lords Capulet and Montague, his motives were pure. Friar Laurence committed a sin through his deception, although his intention was to end the feud and create unity in the two families through love, as we see in his lines:
In one respect I'll thy assistant be;
For this alliance may so happy prove
To turn your households' rancour to pure love. (II.iii.93-95)
However, unfortunately, Friar Laurence's plan of deception backfires, leading to even more deception and finally to Romeo's and Juliet's deaths. Had Friar Laurence been a bit more prudent and much less hasty, he might have found another way to unite Romeo and Juliet in wedlock without the deception. For example, Friar Laurence could have acted as their mediator and confronted the two families on the topic, gaining the families' consent. We know that Lord Capulet already felt a great deal of respect for Romeo since he says as much at the ball. Therefore, other approaches may have been far more successful. Regardless of Friar Laurence's blame in their deaths through his sin of deception, Prince Escalus does not judge him due to his pure motives, as we see when the Prince says in the final scene, "We still have known thee for a holy man" (V.iii.281).
For the setting of the play, sins from the Catholic theology will be used.
Though any act of carnal pleasure committed between Romeo and Juliet were the allowed consumption of their marriage, they lusted for each other . Lust is one of the 7 deadly sins.
'Anyone who looks at a woman lustfully has committed adultery with her in his heart'
-Jesus Christ (Mt 5:28)
Both the Montagues and the Capulets showed animosity towards each other as a result of Pride and Anger other deadly sins.
"Whoever is angry with his brother will be liable to judgment"
"The Devil, the proud spirit, cannot endure to be mocked."
-St. Thomas More, 16th Century
Romeo had sinned when he killed Juliet's cousin who himself killed Mercutio with whom he kept clashing.