There is only one real, meaningful, example of where this has happened, where mathematical theorems were overturned. I am excluding stupid mistakes, like Kempe's mistaken proof of the four color theorem.
The one example is the collection of related proofs that there exists a well-ordering of the reals, a non-measurable set and Banach-Tarsky decompositions, a basis for R as a vector space over Q, a non-principal ultrafilter on the integers, and so on. These theorems were proved using a nonconstructive method in the early 20th century, using the new axiomatic set theory, which was partly constructed to make these types of proof rigorous.
These proofs made a huge stink, because, although they were formally correct in the axiom system, many people still just couldn't believe the object asserted to exist really existed in any Platonic sense of the word regarding the actual continuum as we know it. The fact that the proofs went through in the axiom systems didn't help persuade the skeptics at all, because the axiomatic method included a method to choose uncountably many things simultaneously arbitrarily, and really, the intuition that these were false outweighed the axiomatic system in many people's intuition.
But still, after a half-century of debate, the question was completely settled, as the skeptics died off one by one. By the 1940s and 1950s, the controversial results were just promoted to absolute truth, mostly due to the work of Godel, which showed that the axiom of choice was true in the minimal model constructed to obey these axioms, a model called Godel's L, and therefore was a consistent convention. This result made mathematicians accept the convention, and stop questioning these theorems and accept them as true about the Platonic ideal of the real numbers.
This situation continued for 20 years, until Paul Cohen came along. What Cohen did was to make models in which the axiom of choice failed, and make models in which the continuum was arbitrarily large.
