As an undergraduate I submitted a paper of mine to a reputable journal. Fearing that I would be desk-rejected for simply being an undergraduate, I stated in my submission bio that I was a PhD candidate. It turns out the paper is going to be published and I am curious if there are any long term repercussions or ethical concerns to be worried about on my end. Is it worth issuing a correction or is it statistically unlikely the EIC will even care at this point? Equally as important: would relaying such information result in the paper not being published?
It's maybe too easy to offer advice on such matters when one doesn't have to live with the consequences, but for what it's worth: for your own peace of mind, and to minimize the risk of long term consequences (while possibly increasing the short term risk, though not by much I hope), I think you should send a contrite email to the editor explaining what happened and apologizing. Although there is no telling for sure what the editor would do, considering that your bio status is immaterial to the acceptance decision, I doubt that this will cause them to unaccept the paper. In any case, it's the right thing to do. And if you decide to stay silent and keep your deception a secret, you will have to spend the next few years worrying that it will be discovered, with potentially worse consequences for your career. Nip it in the bud is what I suggest. Good luck!
As a professor who has trained a lot of graduate students (and someone who still recalls what it was like trying to get started in research), I would echo the answers by Dan Romik, Laurent Duval, and Chris John. It might sound like a relatively harmless and justifiable fib, but I guarantee you that if discovered later, it will be seen as a serious blot on your reputation, simply because it suggests that you are prepared to lie to secure a publication. You might never contemplate falsifying results, but people cannot know that and suspicion of your integrity can be fatal to your career.
The good news is, that if you own up before publication, and admit it was an error of judgement, there's a very good chance the editor would be lenient to someone just getting started. Everyone makes mistakes and odds are good he or she was once an eager undergraduate too!
Scientific conclusions in papers may turn untrue, models or data can be wrong. But at the heart of research, intellectual honesty seems to me to be a core value. One should not state something he knows is false. @Dan Romik gave the sounder advice.
Lies are "a little" like bank accounts. Sometimes they get unnoticed in offshore paradise islands for a whole lifetime. Sometimes they get unveiled with compound interests and the necessary prejudice. A little lie at one time turns into tremendous cheating after a few years.
The only situation that could save a little is: would you be a PhD candidate when the paper is published? I believe as @Dan Romik that a status should not be an acceptance reason. But then your bio would be factually correct. And for the sake of honesty, a footnote under your name stating that:
"the author was an undergrad student at the time of submission"
would be nice addition. Indeed, papers written by undergraduates can get noticed, because it can mean early orientation toward research.
Dishonesty like this, if discovered, will cast immediate doubt on the quality of your research. If you are prepared to lie about this what else have you concealed or fabricated?
I would tend to agree that your best bet is to come clean immediately and admit that it was an error of judgement which you now regret. Even if this results in the paper not being published it should at least defuse what might otherwise be a ticking time-bomb for your future career and you can at least truthfully say that you acknowledged and corrected your mistake before you gained anything from it.
Having said that it is not a huge deception and not particularly relevant to the content of the paper. As an undergraduate you probably stand a good chance of being forgiven and having it put down to the inexperience and enthusiasm of youth. Especially if you own up before you get caught.
Get it fixed asap. As others have stated just write to the editor. Even if they reject the paper due to the 'error' this will be the end of it and you'll still have an opportunity to publish in a different journal.
But if this is not corrected then it is not only a problem for your career in academia.
You can't go into politics with something like this over your head.
In Germany, as an example of one jurisdiction, it is a serious offence to misrepresent your title or degrees.
And do not forget about visa trouble. You really do not want to try to explain to an immigration officer why you lied about your degrees on a paper 10 or 20 years ago.
If it is not fixed it does not go away. There is a very permanent record of wrongdoing, how small it may seem.
As a starting point:
Dear Dr. Name-of-Editor,
I was thrilled to receive notification that my submission (title of article) has been accepted for publication. However, I must make one very important correction. When filling out my submission bio, I described myself as a PhD candidate. Actually, I am currently a senior [junior] in the Name-of-Field program at Name-of-University, hoping to begin my PhD studies next fall. I sincerely apologize for the untruth. I hope this does not compromise my submission.
I have explained the situation to my dean of undergraduate studies, Dr. Name-of-Dean. If you would like to contact him/her, here is his/her contact information: (email) (phone).
Instead of the dean, you could provide the name of a mentoring professor.