If the Olive Branch Petition was honored, the American colonists and Great Britain would have continued their relationship. This would mean the following events would have most likely not occurred.
1) American Revolutionary War. - All of the battles that took place after July 1775 (when the Olive Branch Petition was adopted) would have not taken place.
2) Declaration of Independence- America wouldn't have declared themselves an independent nation if they worked out their problems with the British.
3) Articles of Confederation- This constitution was created as a means to unite the American colonists in their fight against the British government/military.