A popular fan theory states that Super Mario Bros. 3 was in actuality just a performance or stage play, as written about by Cracked and Dorkly among others. Is there any basis to this assertment? Has it been confirmed or denied?
Yes, according to series creator Shigeru Miyamoto it was. He nods affirmingly to this very question in this video on Mario myths.
Literary theory is a different sort of thing from a scientific or investigative theory. In most logical fields of reasoning, a theory can be confirmed or repudiated with enough study, investigation, or experimenting. A literary theory, which is closer to what you're talking about here, is a critical interpretation of a piece of literature (and films, plays, and video games all get lumped in as literature).
When we look at a theory like this about a piece of fiction, we can look for hidden meanings and pieces of evidence within the body of the text that support that interpretation. Example: there's a great theory that Nick, the narrator of The Great Gatsby, is gay. Proponents find support in the way he recounts one episode where he loses some clothes in a drunken haze with another male character, the way he admires Gatsby, and the language he uses to describe his female lovers. Opponents, of course, point to the stated fact in the text that he has female lovers. It's a fun debate, and it adds a little replay value (if you'll pardon the phrase) to an old book. Good theories can do that.
But they can't be proven conclusively one way or the other. We're never going to have a solid answer on whether SMB3 was a play or performance of some kind until we can interview Mario about it -- and he's not available. Authors can and do sometimes get involved in discussions like this; JK Rowling speaks to theories on Harry Potter all the time, and sometimes people take the author as an unquestionable authority on the matter. But sometimes the author lies, and sometimes the author just doesn't want to get involved. But even if Miyamoto says something about his fiction, it's totally legit if you read it a different way. If a million people read the Harry Potter books, and the author says later that she meant Dumbledore to be gay, but the million readers didn't interpret it that way -- she doesn't invalidate their experience by saying that. It's just her perspective. Miyamoto's perspective on whether it's a stage play is the same, it's just one guy's perspective. It's not proof; there can't ever be that.
The reality of your experience with the game is really yours to decide. Theories can help you explore it and get deeper, but they can't define what the truth is for you -- that's all about how you read it. It's a discussion, not a verdict.