In the final scene, Romeo appears to be sending Balthasar away from Juliet's tomb because Romeo wants privacy. Romeo threatens Balthasar's life if he interferes with Romeo while he is inside the tomb, saying,
Upon thy life I charge thee,
Whate'er thou hearest or seest, stand aloof
And do not interrupt me in my course. (V.iii.25-27)
Romeo appeases Balthasar by saying that his reasons for wanting to be in the tomb are to see Juliet's face one last time and to take his wedding ring off her finger, however, we know this is a lie.
Romeo's real reason for sending Balthasar away is that he has plans to kill himself with a vile of poison that he has just procured from the apothecary in the first scene of this final act. As we all know, death by poison can actually be quite brutal, even excruciating. Hence, Romeo's real reason for sending Balthasar away from the tomb is that Balthasar may hear Romeo suffering and try to prevent him from committing suicide. Romeo does not want anyone to interfere with his plans for death.
In the final scene of Act Five of Romeo and Juliet, Romeo has traveled to the Capulet's crypt to commit suicide at the deathbed of Juliet, who he mistakenly believes to have died while he was exiled from Verona.
Romeo sends away Balthasar, who has accompanied him to the outside of the crypt, asking that he deliver a letter to Romeo's father. Romeo falsely tells Balthasar that he wishes to enter the crypt alone, claiming:
Why I descend into this bed of death,
Is partly to behold my lady's face;
But chiefly to take thence from her dead finger
A precious ring, a ring that I must use
In dear employment....
In other words, Romeo claims that he wants to look at his beloved's face one last time and to remove her wedding ring from her finger. He threatens Balthasar, telling him that if he dares return to spy on Romeo, he will tear him apart "joint by joint."
For obvious reasons, Balthasar acquiesces to Romeo's demands, and Romeo enters the crypt to act upon his true intentions: taking the poison that he has purchased from the apothecary and dying by Juliet's side.