TITLE: Endless controversy about the correctness of significant papers
QUESTION [156 upvotes]: 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.

REPLY [40 votes]: 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: 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.
FURTHER UPDATE: 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.