Did Friar Laurence rely on logic or emotion when making the decisions to marry Romeo and Juliet, to help Juliet devise a plan to fake her death, and to leave the tomb without Juliet?
To look at the other side of the above argument, one could say Friar Laurence approached the situation of Romeo and Juliet with a logical frame of mind, but things outside of his control interfered. In the first situation, when he decides to marry Romeo and Juliet, he does so to try to end the feud between their families, saying:
"For this alliance may so happy prove
To turn your households' rancor to pure love" (2.3.91-2)
It is not so illogical that this would work. Both the Capulets and Montagues love their respective children and might be more likely to stand by the decision rather than disown them (though Capulet would be a hard sell). Friar Laurence does have an ultimate plan and it is driven by logic.
As for Friar Laurence's second decision to decide to help Juliet fake her own death, he is still acting in accordance with his own internal logic, though the audience is beginning to worry about his judgment. Recall that Juliet has threatened to kill herself rather than marry Paris. In order to save her life (as well as his own reputation), Friar Laurence chooses to do something drastic and he frames it as such:
"I do spy a kind of hope,
Which craves as desperate an execution
As that is desperate which we would prevent" (4.1.70-3)
Friar Laurence knows this plan is dangerous, but he weighs it against the other dangers and considers it the best choice.
For his third decision, to run out of the tomb without Juliet, I have little defense. He seems to let his emotions get the better of him there, with both his fear of the supernatural and his likely fear of judgment from the community for his previous actions.
I would say in a simple and short answer that mostly logic because he thought marrying Romeo and Juliet would end the feud between the two family's
Friar Laurence relied heavily on emotion when making the decisions to marry Romeo and Juliet, plotting Juliet's death, and when he decided to leave the tomb without Juliet.
The Friar did not think through the implications of what would happen if he were to follow through with the decision to follow his heart instead of reason.
Logically, Friar Laurence would not have married Romeo and Juliet as he would know what would happen when the respective families find out what he did. Worst case, Friar Laurence would be without a job, and kicked it of the city.
Had the Friar been thinking logically he would not have married Romeo and Juliet, made a plan to fake Juliets death, and leave the tomb without Juliet.