Edit: a (methodo-)logical proposal to make this thread more transparent
It can be argued that, broadly, there are three quite distinct 'types' of such controversies (and I propose that each answer in here gets tagged, by the respective contributor, if so inclined, by one of the following three tags):
(non-sequitur) This is the nightmare of anyone who has had to referee a long submitted paper and felt the responsibility to make a judgemental statement about the 'Is it true?'-part of a referee's three Littlewoodian responsitbilities: the proof contains many true things, but the goal seems not to be reached, but it is so difficult to justify why one is not convinced, all one can say is 'I am not convinced'.
(propositional-contradiction) By this I mean that the result contradicts, on the coarse boolean level of propositional logic, another published result, and both proofs are long, so ferreting out the error is literally a dilemma, a διλήμματος, with two horns (which most of the time, sadly, are not so easy as to be formalizable in Horn logic). This the dream of anyone who has to refereee a long paper, since then there is an undisputable and documentable reason why one cannot give the go-ahead, if the traditional standards of truth are to be upheld at all (which they should), namely that propositional logic is a conditio-sine-qua-non, something like 'checking an arithmetical calculation modulo two'.
(many-small-gaps) By this I mean that neither (non-sequitur) nor (propositional-contradition) are applicable; the overall line of argumentation is convincing, and, by itself, the claimed conclusion seems credible, too, especially as there is no other proposition proved elsewhere which would propositionally contradict it, but there are lots of small mistakes. This is something between the dream and the nightmare: one can then with good conscious recommend publication, or at least, a second round, but the task of patching up all the small errors still is nightmarishly work-intensive.
None of the above three seems to imply any of the others. On a rough intuitive level, these seem mutually distinct 'types' of controversies around a manuscript (in my experience).
I'll 'tag' my proposed contribution to this thread with the second-named 'type'.
A proposed contribution to this thread.
(propositional-contradiction) With trepidation (since I am only beginning to understand what the real issues are), and due respect, let me mention one of the most famous examples these days. To repeat myself: I know that there are many many others round here whom it would behoove more to mention this.
Endlessly fascinatingly- and fertilely-controversial is:
M. M. Kapranov, V. A. Voevodsky: ∞-groupoids and homotopy types.
Cahiers de Topologie et Géométrie Différentielle Catégoriques (1991)
Volume: 32, Issue: 1, page 29-46
Now the question is of course whether this qualifies as 'endless controversy' since even one of the authors readily acknowledged that there was an error, but a fruitful error, indicative of the traditional methods (both the formal-methods and the social-methods) being inadequate to give 'durable wings' with which to do the 'flights of fancy' (in a positive sense) of higher category theory.
But, while still learning some of the relevant subject matter (and, myself, being mostly working to understand the comparably humble example of the unambiguous interpretability of pasting schemes in good-old-bicategories), I think I can recognize that the above example satisfies each of the requirements
famous (why? look around...)
endless (why? since this dedicated MO thread seems so unconclusive (to me); after as yet 2624 views on a professional focused site, said thread contains only a "guess" and detailed confirmation *that there is an incorrectness in the sense of propositional logic but it still seems not clear (to me) how to pin down the reason for why the authors 'went wrong'.
controversial (why? since one of the authors himself in public lectures said that at first he did not take Simpson's statement that something was wrong serious, rather thought that it was wrong to state that something was wrong; what is endlessly fascinating about this example is the expressiveness of the mathematics which gave rise to this 'controversy')
significant (why? because, similar to e.g. Poincaré fertile errors in 'Analysis Situs' and the 5 subsequent 'patches', Kapranov-Voevodsky's error turned out to be a fertile error, for example by motivating one of the authors to find an alternative formal system for mathematics)
A micro-summary is given on a page hosted by the Institute of Advanced Study in Princeton:
During these lectures, Voevodsky identified a mistake in the proof of a key lemma in his paper. Around the same time, another mathematician claimed that the main result of Kapranov and Voevodsky’s “∞-groupoids” paper could not be true, a flaw that Voevodsky confirmed fifteen years later. Examples of mathematical errors in his work and the work of other mathematicians became a growing concern for Voevodsky, especially as he began working in a new area of research that he called 2-theories, which involved discovering new higher-dimensional structures that were not direct extensions of those in lower dimensions. “Who would ensure that I did not forget something and did not make a mistake, if even the mistakes in much more simple arguments take years to uncover?” asked Voevodsky in a public lecture he gave at the Institute on the origins and motivations of his work on univalent foundations.
Voevodsky determined that he needed to use computers to verify his abstract, logical, and mathematical constructions. The primary challenge, according to Voevodsky, was that the received foundations of mathematics (based on set theory) were far removed from the actual practice of mathematicians, so that proof verifications based on them would be useless.
fifteen years later
seems to approximate "endless" rather closely.
Again, my apologies if this is off-topic for some reason that I do not see, and I know it is debatable whether this counts as endless controversy, maybe indefinite fertility would be a more fitting heading for this example.