Romeo's lack of moderation can be seen all throughout the play. However, it's really his lack of moderation and forethought prior to the start of his relationship with Juliet that leads to their deaths.
His lack of moderation is first seen in his excessive emotional response to being rejected by Rosaline. While feeling blue over love lost is normal, Romeo has taken his feelings to such an extreme that he has been seen staying out all night long, night after night, and is seen each morning at dawn in a certain part of town, under a grove of trees, crying. Romeo's father is so concerned by his behavior that his father fears he may do himself harm if no one can break through to him to offer some advice. Benvolio begs Romeo to forget about Rosaline, but he refuses to listen to his advice, showing us that Romeo allows himself to be governed by his immoderate emotions rather than by his rational mind.
It's this same emotional drive and lack of rational thought that leads him to be persuaded against his better judgement to crash the Capulet's ball. It is really the moment he crashes the Capulet ball and is recognized by Tybalt that seals his fate for early death and Juliet's as well because, had this never happened, Tybalt never would have felt insulted by Romeo and challenged him to a duel, which lead to Tybalt's death, Romeo's banishment, and eventually Romeo's death as well. We know that crashing the ball was against his better judgement because he says as much. He states he had a dream he believed was prophesying that "this night's revels" would lead to his own "untimely death" (I.iv. 113-18). Yet, he next tells his friends to lead him on regardless, showing us that he is letting himself be guided by his emotions rather than his rational thoughts and rational hesitations. Hence, allowing himself to be influenced by his friends into crashing the ball shows us that he lacks both moderation in emotions and forethought, and it is this key event that leads to his death, just as he prophecies.
However, there is one moment in which he could have saved both his and Juliet's lives had he only been more rational and moderate in his emotions, even after the death of Tybalt and his exile. Had he been more rational when he saw Juliet in the tomb, he would have recognized that she was not truly dead, which would have spared him from poisoning himself and saved Juliet from suicide. When Romeo finally beholds Juliet in the tomb, he notices that her lips and cheeks still look rosy, as we see in his important lines:
Beauty's ensign yet
Is crimson in thy lips and in thy cheeks,
And death's pale flag is not advanced there. (V.iii.94-96)
It is his immoderate passion for Juliet and her beauty that is making him think this way. Had he been using his more rational mind, he would have noted that it is impossible for Juliet to be both dead and still rosy. Had he realized that, Friar Laurence's plan would have worked out accordingly and the story would have had a happy ending.
Hence we see that it is ultimately Romeo's lack of moderation in his emotions, lack of reason, and lack of forethought that eventually leads to his and Juliet's deaths.
The obvious one would be when Romeo kills Tybalt. He kills him in a fit of anger as a result of the death of Mercutio. There is no forethought and it is what gets Romeo banished from the land.
But the one that isn’t so obvious is the death of Paris at the end of the play. Have you ever seen either the Zeffirelli or Luhrmann version of “Romeo and Juliet”? If you have, then you’ll notice that the one crucial part they BOTH leave out is the scene where Romeo kills Paris. The reason why it’s left out is because Paris’ death at the hands of Romeo complicates the entire idea of Romeo standing in as a figure for love and romance in a period of turmoil. He kills Paris simply because he doesn’t want to get caught because, after all, Romeo is a criminal and a fugitive. It’s selfish and he does it without weighing out the consequences of his actions. Needless to say, Paris’ death is unnecessary and certainly does complicate the relationship with Juliet (even in death!). The implication is that love is not lollipops and puppies, but something that will make you do things without moderation and forethought. Romeo has become everyone he despises.
Paris’ death puts everything you know about the play (its general themes of love, relationships, and solidarity) into question. Not just between characters, but the message the play is trying to communicate.