The Red Sash of Tokugawa Ieyasu
When Brandy entered the Shrine of the Silver Monkey, the door leading to the Room of the Secret Password was already open, but Brandy ignored it and completed the monkey anyway. I wonder why the door was open. Could this be a production error? And why did she complete the monkey anyway? Just go through the shrine and into the other room.
When she completed the Monkey, the door to the Quicksand Bog opened. And yes, I think it's a production error.
The Secret Password-Pharaoh's Secret Passage door being open without needing to complete an objective in the Smashed Printing Plate run would count, also.
I kind of disagree, I think the producers did that intentionally sometimes in order to lower the difficulty in the layout, if that makes sense at all.
For example, Brandy was not required to assemble the monkey at all and could've ran into the Room of the Secret Password, bypassing the Shrine altogether. However, if she decided to assemble the monkey, then the Quicksand Bog door would've opened, allowing her to avoid the Room of the Secret Password. Notice that both the Secret Password and Quicksand Bog had a temple guards, meaning that had she took the Secret Password path, she would've met both temple guards.
And its not the only run with a completed objective either. The Stolen Arm of Shiva, The Mask of Shaka Zulu, Henry VIII's Great Seal, The Walking Stick of Harriet Tubman, The Smashed Printing Plate of Frederick Douglass, and The War Fan of the 47 Ronin all had some sort of completed or half completed objective during the run.