Endless controversy about the correctness of significant papers In principle, a mathematical paper should be complete and correct. New statements should be supported by appropriate proofs. But this is only theory. Because we often cannot enter into the smallest details, we "prove" wrong statements here and then. I plead guilty, having published myself one or two false (fortunately minor) papers.
So far, this is not harmful. The research community is able to point out incorrect statements, at least among those which have some importance in the development of mathematics. In time, the errors are fixed; this is the role of monographs to present a universally accepted state of the art of a topic. 
But sometimes, hopefully rarely, the technicalities are such that a consensus does not emerge and a controversy raises, between the author and their critics. I have an example in the realm of wave stability in PDE models for fluid dynamics. The controversy has lasted for a decade or two and I don't see how it can be resolved some day; it could just kill the topic.

Are there famous endless controversies about the correctness of a significant paper? Are there some significant mathematical questions, that remain unsettled because people disagree on the status of released proofs? What should we do in order to salvage mathematical topics that suffer such tensions?

In this question, I am not concerned with other kinds of controversy, about priority or citations.
 A: Nearly two decades after the publication of Adam Elga, Self-locating belief and the Sleeping Beauty problem, Analysis, 60(2): 143-147, 2000, there is apparently no consensus as to the resolution of the problem. See Peter Winkler,  The Sleeping Beauty Controversy, The American Mathematical Monthly, Vol. 124, No. 7 (August-September 2017), pp. 579-587
A: (Also mentioned in Oliver Nash's comment) 
From a February 2017 article in Quanta Magazine called "A fight to fix geometry's foundations" (the original has relevant links in the text):

...in 2012, a pair of researchers — Dusa McDuff, a prominent symplectic
  geometer at Barnard College and author of a pair of canonical
  textbooks in the field, and Katrin Wehrheim, a mathematician now at
  the University of California, Berkeley — began publishing papers that
  called attention to the problems, including some in McDuff’s own
  previous work. Most notably, they raised pointed questions about the
  accuracy of a difficult, important paper by Kenji Fukaya, a
  mathematician now at Stony Brook University, and his co-author, Kaoru
  Ono of Kyoto University, that was first posted in 1996.
This critique of Fukaya’s work — and the
  attention McDuff and Wehrheim have drawn to symplectic geometry’s
  shaky foundations in general — has created significant controversy in
  the field. Tensions arose between McDuff and Wehrheim on one side and
  Fukaya on the other about the seriousness of the errors in his work,
  and who should get credit for fixing them.
More broadly, the controversy highlights the uncomfortable nature of
  pointing out problems that many mathematicians preferred to ignore. “A
  lot of people sort of knew things weren’t right,” McDuff said,
  referring to errors in a number of important papers. “They can say,
  ‘It doesn’t really matter, things will work out, enough [of the
  foundation] is right, surely something is right.’ But when you got
  down to it, we couldn’t find anything that was absolutely right.”

A: The Jordan curve theorem asserts that a simple continuous closed curve separates the plane into two distinct connected open sets.
Whether Camille Jordan's original proof is correct or not seems to be a subject of controversy even today. 
Here is a 2007 article by Thomas Hales that "defends Jordan’s original proof of the Jordan curve theorem" (from the abstract) and contains opinions from mathematicians who have pondered on the matter. 
Here is an excerpt of the paper.

In view of the heavy criticism of Jordan’s proof, I was surprised when
  I sat down to read his proof to find nothing objectionable about it.
  Since then, I have contacted a number of the authors who have
  criticized Jordan, and each case the author has admitted to having no
  direct knowledge of an error in Jordan’s proof. It seems that there is
  no one still alive with a direct knowledge of the error.

You can find the proof of Jordan in his cours de l'ecole polytechnique p92ff if you want to form your own opinion. 
Don't forget to contribute your own short proof of the Jordan curve theorem, and receive my applause for adding to the long list of proofs of the Jordan theorem, which are probably not short enough, and that contain an argument that is completely trivial (Hales), unsatisfactory to many mathematicians (Veblen), essentially correct (Reeken), invalid (Courant and Robbins), incorrect (M. Kline), not sufficient (another Kline) or simply in need of additional details (Mathoverflow).
A: 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.

The

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.
A: Stanley Yao Xiao's comment has been upvoted so highly that it seems worth posting as an answer.
There is a currently unresolved controversy over Shinichi Mochizuki's claimed proof of the abc conjecture.  In a ten-page note, Peter Scholze and Jakob Stix have stated:

We, the authors of this note, came to the conclusion that there is no proof. We are going to explain where, in our opinion, the suggested proof has a problem, a problem so severe that in our opinion small modifications will not rescue the proof strategy.

Mochizuki, however, maintains that there are no problems with his proof and that Scholze and Stix suffer from "fundamental misunderstandings."
UPDATE 1: Mochizuki's work has been accepted for publication in PRIMS, even though most number theorists believe that the proof is incomplete. For more details, see this April 2020 post on Peter Woit's blog.
UPDATE 2: Mochizuki's papers were published in PRIMS in March 2021.  However, the publication of the papers has not ended the controversy; see the March 2021 post on Peter Woit's blog.
UPDATE 3: NHK aired a documentary (in Japanese) in April 2022.  It tried to present "both sides of the story," though according to the reviews I've seen, it presented Mochizuki's theory as being "too difficult to understand" rather than having a huge gap. Again, there is some additional commentary on Woit's blog. In other news, there is a guest post by Kirti Joshi on David Roberts's blog which, while not resolving any of the big issues, presents some work that Joshi says "provides new evidence regarding Mochizuki’s work."
A: As far as I know, Wu-Yi Hsiang still maintains that his proof of the Kepler conjecture is complete and correct.  Perhaps this does not quite meet your criteria because it seems that nobody other than Hsiang believes that his proof is complete and correct; is that enough agreement to use the word "consensus"?
In the introduction to his delightful anthology, 18 Unconventional Essays on the Nature of Mathematics, Reuben Hersh states, regarding the Flyspeck project of Thomas Hales, that he does not know anyone who either believes that the project will be completed or that, even if claimed to be complete, it will be universally accepted as definitively verifying the correctness of the proof.  Of course, the completion of Flyspeck was announced in 2014.  My impression is that most people accept that Flyspeck has settled the Kepler conjecture, but there are probably still some skeptics.  Back in 2008, I had an email exchange with someone who pointed out that HOL Light is based on OCAML, and that the formal correctness of OCAML has not been established.  It may be that there will always be a nontrivial minority of mathematicians who remain skeptical of results whose only proof is not "surveyable" or "humanly comprehensible," in which case such results may remain permanently controversial.  (One could imagine that one day the Kepler conjecture will have a humanly comprehensible proof, but there will surely be other results, e.g., in extremal combinatorics, that are unlikely to have any proof other than an immense computer verification.)
A: F. Enriques' claimed in 1904 that, given a smooth projective surface $S$ with irregularity $q>0$, what we call nowadays the Picard scheme of $S$ is an abelian variety of dimension $q$.
Enriques' algebraic proof was considered controversal and led to many disputes among the geometers of the Italian school (in fact, they called Enriques' claimed result the fundamental theorem in the theory of irregular algebraic surfaces, or also the theorem of completeness of the characteristic series).
Actually, Enriques' proof contained lacunae, as well as the subsequent proof by Severi. The first correct proof, using transcendental methods, was given by Poincaré in 1910. For a correct algebraic proof in characteristic $0$ it was necessary to wait for the work of Grothendieck, 50 years later.
Today it is known that the result is false in positive characteristic, as shown by Igusa in 1955. In fact, he constructed a smooth projective surface $S$ with $\mathrm{Pic}^0(S)$ non reduced, and hence not an abelian variety.       
A: One classic example of this, though now resolved, is Euler's polyhedron formula $V - E + F = 2$. The formula initially was asserted without qualification, and gradually people began to point out cautiously (I suppose in deference to Euler) that there were "exceptions" to the theorem. The history of this formula is explored in Imre Lakatos's lovely book Proofs and refutations, written in the form of a drama, but with copious historical references. 
This was a fertile error in the sense given above by Peter Heinig.
A: A seemingly endless controversy that has already lasted 15 years surrounds Yaroslav Sergeyev's proposed theory of infinity, first published in 2003.  He has over twenty publications on this topic.  Critics maintain that his theory is inconsistent, with whatever valid material actually derived from earlier rigorous sources.  He also has some backers including one that posted an answer at MO.
Complete disclosure: I am one of the authors of this 2017 publication in Foundations of Science critical of Sergeyev's production.
Some other possibilities are mentioned in this question.
