The lives of both Juliet and Romeo are affected adversely by Paris. Because of her forced betrothal to Paris by her parents in Act IV, Juliet, who has already secretly married Romeo, resorts to a desperate act in order to prevent herself from committing bigamy, and later at the Capulet tomb, Romeo, driven to fury by Paris's accusations that he is the cause of Juliet's suicide over the loss of Tybalt, kills Paris.
In a sense, then, Paris is the fulcrum of the tragic ends of the two lovers. In the first act, when Lady Capulet introduces the idea of her marriage to Paris, Juliet tells her mother that she will "look to like, if looking liking move"--in other words, she will look if at Paris and try to like him (1.3.99). This urging by her mother plants the idea of love and marriage in the mind of the young Juliet, who may not have considered such concepts until this time. At any rate, this mention of Paris certainly foreshadows his involvement.
Certainly, Lord Capulet's vehement insistence that Juliet marry Paris in Act IV leads Juliet, whose emotions run high over the death of Tybalt and banishment of Romeo, to become extremely agitated. In this highly emotional state, Juliet hurries to Friar Laurence, who fashions a desperate plan to prevent Juliet from committing suicide because she has told him,
Oh, bid me leap, rather than marry Paris
From off the battlements of yonder tower....
To have an unstained wife to my sweet love. (4.2)
The priest gives her a sleeping potion which will create the semblance of death in Juliet for a time, during which he may be able to reason with Lord and Lady Capulet after explaining that Juliet has already been married.
Thus, the possibility of marriage to Paris indirectly places Juliet in a tomb, presumably dead. Further, should Romeo have not assumed that she were dead, he would not have come to her tomb where he encounters Paris. Moreover, there would have been no reason for Romeo to have wished to kill himself, which then leads Juliet to wish to join him in death after she revives from the potion.