Was there any time in The Big Bang Theory - whether literal or referenced, that Dr. Sheldon Cooper was proven wrong by someone?
Apparently, there were a couple of episodes where he was wrong. I am adding what I remember, but will add more if I come across them later.
In the episode, The Hamburger Postulate, Dr. Leslie Winkles corrects his wrong equation.
Sheldon: My equations, someone’s tampered with my equations.
Leonard: Are you sure?
Sheldon: Of course I’m sure. Look at the beta-function of quantum chrono-dynamics, the sign’s been changed.
Leonard: Oh yeah. But doesn’t that fix the problem you’ve been having?
Sheldon: Are you insane? Are you out of your mind? Are you…. hey look, that fixes the problem I’ve been having.
Lesley: (entering) You’re welcome.
Sheldon: You did this?
Lesley: Yeah, I noticed it when I got up to get a glass of water, so I fixed it, now you can show that quarks are asymptotically free at high energies. Pretty cool, huh?
In the episode, The Bat Jar Conjecture, when Sheldon is participating the Physics Bowl against Leonard and his team, both parties are stuck on the last question. The janitor on Sheldon's team answers it right, but Sheldon refuses to make it the team's answer due to his ego.
Sheldon: Hang on, hang on a second, that’s not our answer. What are you doing?
Third Floor Janitor: Answering question. Winning physics bowl.
Sheldon: How do you know anything about physics?
Third Floor Janitor: Here I am janitor, in former Soviet Union I am physicist. Leningrad Politechnica. Go Polar Bears.
Sheldon: Well that’s a delightful little story, but our arrangement was that you sit here and not say anything, I answer the questions.
Third Floor Janitor: You didn’t answer question.
Gablehouser: Denied. I need your official answer.
Sheldon: No. I decline to provide one.
Gablehouser: Well, that’s too bad, because the answer your teammate gave was correct.
In the episode The Hawking Excitation, when Hawking calls him to discuss about his paper, Hawking tells him that he made a arithmetical error on page 2. After that, he faints.
Hawking: That’s nice. Too bad it’s wrong.
Sheldon: What do you mean wrong?
Hawking: You made an arithmetic mistake on page two. It was quite the boner.
Sheldon: No, no, th-th-th-that can’t be right. I-I don’t make arithmetic mistakes.
Hawking: Are you saying I do?
Sheldon: Oh, no, no, no, of course not. It’s just, I was thinking… Oh, gosh, golly, I made a boo-boo, and I gave it to Stephen Hawking.
Hawking: Great, another fainter.
In the episode, The Jiminy Conjecture, Howard and Sheldon bet whether the cricket is ordinary field one or snowy tree one. Apparently, Sheldon was proven wrong about the cricket being snowy tree one.
Prof Crawley: It’s a field cricket.
Sheldon: No, no, wait. Dr. Crawley, are you sure?
Prof Crawley: Young man, I’ve been studying insects since I was eight years old. You know what they used to call me in school? Creepy Crawley.
In The Romance Resonance he read a data table in a text book wrong
Sheldon: This is the very copy of The Handbook of Chemistry and Physics in which I looked up the reaction rates of mendelevium and…
Amy: And what?
Sheldon: No. No, no, no, no.
Amy: What’s wrong?
Sheldon: I’ve made a horrible mistake.
Amy: What are you talking about?
Sheldon: This table, it’s in square centimetres. I read it as square metres. You know what that means?
Amy: That Americans can’t handle the metric system?
Sheldon: Amy, I was off by a factor of 10,000.
Sheldon has to draw "polish" in a game of Pictionary. He draws several pictures that make reference to Polish things and has to explain them all when no-one correctly guesses.
Penny then points out: Excuse me, the word is "polish." See? Small "P."
Sheldon: Ah! So it is. I guess we both share blame on this one.
Sheldon has committed the team to completing the prototype of Howard's guidance system in a unrealistic timeframe of 2 months. Once they start work on the project, Sheldon realises he's made a mistake and starts to get nervous and distracting to Howard and Leonard who both want to knuckle down:
Sheldon: You know, the interesting fact about the rectum...
Leonard: We are dealing with an impossible deadline from the Air Force because of you. So have an energy drink, don't have an energy drink. Order suppositories and shove 'em wherever you want, I don't care!
Sheldon: You don't shove them. They come with an easy-glide applicator.
Leonard: Right. Listen to me. We can't do anything until you do your part. So get up in front of this whiteboard and do it!
Sheldon: I can't.
Leonard: Yes, you can.
Sheldon: No I can't figure out the math. I've been racking my brain for days, and I've got nothing.
Sheldon: I can't do it. I'm not as smart as I think I am. I'm so sorry. This is all my fault.
In “The Hofstadter Isotope,” S2E20, Sheldon is ironically schooled by comic book store entrepreneur Stuart on a key point related to the philosophy of science:
Stuart: Oh, Sheldon, I'm afraid you couldn't be more wrong.
Sheldon: More wrong? Wrong is an absolute state and not subject to gradation.
Stuart: Of course it is. It's a little wrong to say a tomato is a vegetable; it's very wrong to say it's a suspension bridge.
Sheldon has no response, as he realizes he’s now wrong a second time - this time about the very nature of being wrong. As a theoretical physicist, Sheldon knows for example that Newtonian physics and general relativity are both extremely accurate models of gravity in their respective domains, but they’re not perfect and thus in some sense both “wrong.” But GR makes more accurate predictions in extreme conditions, and so is “less wrong.” And both theories are less wrong than Aristotle’s notions that heavier bodies fall faster than lighter ones, and that falling is a “natural motion” returning objects to their proper positions. Science is never 100% right; that’s not a failing of science, it’s the lifeblood.
There is a scene, taken from Season 3, Episode 4, "The Pirate Solution", where Sheldon was "wrong" twice (thanks @Fabio Turati).
First was an equation when working with Raj, who was hired to work "for" Sheldon in the University. They have an argument about the correctness of a formula that Sheldon created, and Raj proves him wrong, while correcting the equation.
Shortly after the argument, Raj exclaims "Can you do this?", and does a magic finger trick, which Sheldon cannot replicate.
The scene is shown below:
In The Cooper/Kripke Inversion (S06E14), the university forces Sheldon and Barry Kripke to work together. Sheldon is none too pleased with having to work with Kripke as a team of equals, Kripke being someone that Sheldon not only sees as far more inferior in intellect and ability than himself (just as he does with most other people) but someone he also despises as a person.
Sheldon looks forward to reviewing (and presumably mocking) Kripke's research, expecting it to be unimaginative and of poor quality, so is quite upset when he finally gets a copy.
Later in the day, sitting at home with Amy:
Amy: Is everything okay?
Sheldon: I'm fine.
Amy: All right, well, how was work today? Did you exchange your research with Kripke?
Amy: Sheldon, what's going on?
Sheldon: I read his research, and it's leaps and bounds ahead of mine. Which means the mommy of the smartest physicist at the university is not my mommy as I had thought. It's his mommy.
Running theme in the series has been Sheldon's disrespect towards engineers (he's referred to them as the Oompa loompas of science) and towards Howard in particular, whose occupation and intellect he has repeatedly mocked.
Sheldon is promoted to Professor at the university, a role which means he has to teach classes. However, he is a little put out when he finds out that no one wants to take his class because of his reputation for being obnoxious.
Howard offers to sign up:
Howard: Hey, what if I took your class?
Leonard: Why would you do that?
Raj: Yeah, why would you do that? What's wrong with you?
Howard: I'm thinking about getting my doctorate, and he wants to teach-- why not?
Sheldon: Oh, Howard. I appreciate the gesture, but this is a graduate-level physics class. I don't think you'd understand a single thing I was talking about.
Leonard: Ask why not again; I've got an answer.
Howard: Sheldon, I'm more than smart enough to take your class.
Sheldon: (pauses then chuckles, mockingly): No.
Sheldon: How would you determine the ground state of a quantum system with no exact solution?
Howard: I would guess a wave-function and then vary its parameters until I found the lowest energy solution.
Sheldon (impressed): Hmm.
Sheldon: Do you know how to integrate X squared times E to the minus X, without looking it up?
Howard: I'd use Feynman's trick-- differentiate under the integral sign.
Sheldon (more impressed): Okay.
Sheldon: Um What is the correct interpretation of quantum mechanics?
Howard: Since every interpretation gives exactly the same answer to every measurement, they are all equally correct. However, I know you believe in the Many Worlds Interpretation, so I'll say that. Now do you think I'm smart enough?
Sheldon: (pauses then chuckles, mockingly): No.
Although Sheldon is not definitively proven wrong in his belief that Howard is not "smart", he clearly expected that Howard would not be smart enough to answer the questions posed and did not want to concede that, by getting the questions right, Howard is actually much smarter then he had been giving him credit for.
Sheldon: Penny. I am a physicist. I have a working knowledge of the universe and everything that's in it.
Penny: Who's Radiohead?
Sheldon: (dumbfounded look for a few seconds) I have a working knowledge of the important things in the universe.
Now, this isn't an admission of being wrong but it disproved his remark about knowing everything in the universe. Only after Penny's snarky question did Sheldon move the goalposts.