EDIT: The episode I had in mind turns out to be the work of Robert Coleman repairing a gap in a paper by Manin about the Mordell conjecture over function fields. See comments by KConrad below, giving specific references. Note that this is not about a false result, it is about an accepted proof with a gap that was found 20 to 25 years later and repaired.
Original:Requesting assistance with a memory.
This being community wiki, I will give my vague memory. I think someone who was actually there could tell a good story. I have been searching with combinations of words on google with no success.
Anyhoo, when I was in graduate school at Berkeley in the 1980's, a professor, whom I think was likely Robert Coleman, told us a story about a celebrated result on "function fields"
or the function field version of something...
The accepted proof was by someone really big, on google I kept running across the name Manin but I am not at all sure about the name. Prof. Coleman decided to present the proof to a class/seminar. Partway through it became clear that the accepted proof just did not work. I have a sense that the class and professor were able to clean up the proof but I have no idea what publication may have come of this. There is also the chance that the seminar did not occur at Berkeley, rather at an earlier job of the professor concerned. Sigh.
So, there are a few ways this story could be filled in. Many MO people are students or postdocs at Berkeley, somebody could walk down the hall and ask Prof. Coleman if that was really him, and if so what actually happened, or ask Ken Ribet, etc.. Again, someone on MO with encyclopedic knowledge of every possible use of the phrase "function field" might be able to say. Or someone very old, yea, verily stricken in years, like unto me.
Finally, note that the title and text of the OP's question disagree a little, and people have posted both "results" that remained false and correct results with incorrect proofs. Also, my memory is really quite good, but I heard this story once and did my dissertation on differential geometry and minimal submanifolds.