Does Friar Laurence rely on logic or emotion when making the decision to leave the tomb without Juliet? What are arguments for both sides?

I would argue it is emotion that drives Friar Lawrence to leave the tomb, much as emotion drove his hasty actions throughout the play. He does not want his plan found out, and so first tries to plead with Juliet to leave so they can evade capture.  When Juliet refuses, he exits quite quickly (admitting later that a noise scared him away). Fear of being caught makes him leave, but then guilt brings him back to admit his role in all this to the Prince. Responding to both fear and guilt implies his emotions are driving his decision making. 

But if you were to argue that logic was driving his actions, you could make the case that it's quite calculated how he starts to devise how he can cover up his actions. By depositing Juliet in a convent and not being found at the scene of the crime, no one else would be the wiser of his part in the tragedy and no direct link to the Friar could ever been proven.