Without Friar Lawrence, perhaps no one else would have agreed to marry Romeo and Juliet; he only does it because he hopes that it will bring peace between the two families. It is likely that a different friar would not have felt the same way. Further, another friar would likely not have counseled Juliet to fake her own death, deceive her family, and run away. It's a pretty sneaky thing for a man of God to suggest, so it seems entirely possible that another friar would not have advised it, and thus Romeo would never have believed she was dead and killed himself. Moreover, if Paris did not exist, Juliet would likely not have been forced into a marriage so soon. Her father said, in the beginning, that he thought she was too young to marry; it was only Tybalt's death and Paris's readiness that compelled him to move forward with the wedding plans so quickly. Therefore, without a willing groom in Paris, Juliet would probably not have been urged to marry and would never have felt the desperation that causes her to fake her death.
Thus, it seems very probable that, without either character, the end would have been different. Without Friar Lawrence to marry the couple or suggest that Juliet fake her death (which was meant to fool her parents but ended up fooling Romeo as well), it seems less likely that the play would have ended with the lovers' suicides. Similarly, without Paris's waiting arms, Capulet would have had no one in particular to whom he could marry Juliet after Tybalt's death, and so she would not have been made desperate by the idea of having to marry again when she was devoted to her first husband. Therefore, Romeo and Juliet would have, at least, had more time to figure out a plan, to take their time and make sure all communication was complete, and this would have prevented Romeo's misunderstanding; it seems unlikely that they would have killed themselves in this case either.