Say I ask a question. It then gets marked as a duplicate (possible or exact) by someone commenting so. However, the question is not a duplicate, the person who posted the comment either didn't understand the question or just looked at the title. What would I do to get rid of the box that says "Possible duplicate of (link)" at the top of my question?
Unfortunately for you, there is no way to remove that message. But don't worry, only you can see it for now. Until the question actually gets closed as a duplicate, nobody else will see that message. Should nobody cast the same vote, then the close vote will expire at some point, taking the message with it.
As stated by Anna Lear, you can use this opportunity to clarify your question and make it really clear that it's not a duplicate. Don't just say it isn't one, but inform us how it differs.
Should the question then inadvertently get closed as a duplicate, you'll already have prepared your rebuttal. Either the community can vote to reopen it based on your clarifications. (The preferred route. And you can cast the first vote if your rep is >250) Or you could at some point flag it for moderator attention and have them take a look at it. Use the "other" flag option and explain what the situation is and how your question was closed incorrectly.