Shakespeare's reason for including both the deaths of Paris and Lady Montague was to further prove his point that violent, uncontrolled, passionate emotions lead to dire consequences, including destruction, such as death.
Not only does Paris's death serve to represent the consequences of Romeo's uncontrolled, violent, passionate emotions, his death also represents the consequences of Juliet's own uncontrolled, passionate emotions as well. We actually learn a great deal from Shakespeare's inclusion of Paris in the final scene. We get to witness Paris severely mourn the loss of Juliet, which tells us that he truly does love her dearly. He even loves her as dearly as Romeo. We especially see how deeply Paris cares for Juliet when we see him promise to nightly water the flowers he is sprinkling on her "bridal bed," meaning her grave, either with "dew" or with his own "tears distill'd by moans" (V.iii.12-15). In other words, we see that Paris is completely heartbroken over Juliet's death and in deep mourning. This is important because it gives us the chance to question whether or not Juliet actually made the correct decision in choosing Romeo over Paris. Juliet's decision was completely guided by impetuous emotionalism and uncontrolled passion. Had she allowed herself to be guided by rational thought she might have seen that Paris has many qualities above Romeo: He is wealthier; he is more mature; and he does not give into rash, impetuous inclinations that are purely driven by violent, passionate emotions as Romeo does. However, unfortunately, Juliet does choose Romeo and the consequence of her choice is early death. Hence, through placing Paris at Juliet's grave, which places him in a position to be killed by Romeo, Shakespeare is allowing the reader to further see the consequences of violent, uncontrolled, passionate emotions, such as Juliet's.
Not only do we see Paris mourn Juliet at her tomb, we see him be killed by Romeo at her tomb. In fact, Romeo begs Paris to leave him alone and not provoke him to wrath, forcing him to kill Paris as well as Tybalt. Plus, Romeo blames himself for Juliet's death, thinking that she died of a broken heart due to his banishment. We see him begging Paris in his lines:
Think upon these gone;
Let them affright thee. I beseech thee, youth,
Put not another sin upon my head
By urging me to fury. (60-63)
Romeo is correct in believing that several deaths were his fault. Tybalt fled the scene after killing Mercutio giving Romeo the chance to make a choice. Romeo could have made the choice to let the law punish Tybalt rather than deciding to avenge his best friend himself. Romeo's decision for revenge was driven by his rash emotionalism, particularly his uncontrolled, violent, passionate emotions. Not only that, Romeo's decision led to other deaths, namely, his own, Juliet's, and even Paris's. Hence, in having Romeo kill Paris Shakespeare was once again demonstrating the dire consequences associated with uncontrolled, violent, passionate, emotions.
Lady Montagues' death is also further proof of Shakespeare's moral. Lady Montague died of grief over Romeo's exile. Romeo's exile was the consequence of his own decision but also of his father's decision, as well as Lord Capulet's, to continue the family feud. Therefore, Lady Montague's death is further proof of Shakespeare's moral concerning violent, passionate emotions.
There are a couple of reasons the deaths of Paris and Lady Montague are included in this play. Some are literary and some are not quite. First of all, Romeo and Juliet is a tragedy, and in a tragedy the world is "out of order," so to speak. To restore balance to the fictional world of the play, many characters that are problematic must die. The more serious the disturbance to the "natural order" of things, the more characthers must die. (Witness the almost total devastation at the end of Hamlet.)
Also, we often make the mistake of thinking of Shakespeare as sophisticated and "high brow". The truth is, at the time he was living, these plays were popular entertainment more in the vein of many movies we see today. For instance, if you're going to see Friday the 13th, you expect, and actually want to be, scared. If you were going to see a tragedy in the 16th century, you expected and wanted to see sorrow, blood, and death. Shakespeare knew now to give the people what they wanted.