How important are proofs in physics? If something is mathematically proven to follow from something we know is true, does it still require experimental verification? Are there examples of things that have been mathematically proven to some reasonable degree of rigor (eg satisfy a mathematician) that turned out to be false based on experiment?
Mathematical proofs relate to exactly how a MODEL will behave. They don't have much to do with how the real world behaves. If the mathematics is carried out correctly, then one has "proven" how the model will behave.
The reason for experimentation, is to find out if the completely fictional MODEL that somebody simply made up, behaves in any way the same, as observations suggest the real world behaves. If it doesn't, that is not an indication of a mathematical error, it simply means the fictional model is not a good description of the real world, and must be changed.
Mathematical proof is to physics roughly what syllogism (or some other fundamental inference rule) is to logic. Namely, it begins from assumptions modelling our conception of some physical reality and shows what must be so if the assumptions hold, but it cannot say anything about the underlying assumptions themselves. A simple example was given by dmckee in his comment:
One must always test results from physics "mathematical proofs" with experiment. Indeed it could be argued that the building of such mathematical proofs is the main job done by theoretical physicists, and the sole reason for building them is to discover what the theory in question foretells that is falsifiable (see Wiki page on falsifiability). There are two "experiments" that need to be done on such a "proof":
So, in short, the process of experimental verification is what sets mathematics and physics apart.
Notice that even if the proof "fails" in step 2, it has been invaluable to physics because it then becomes a proof by contradiction, i.e. it tells us that one or more assumptions underlying the proof must be wrong and therefore our ideas about what really is going on in experiment need to be revised. The proof itself if shown to be such by step 1 above cannot be "wrong".
To answer your question:
As discussed in step 1. above, there are no "mathematical proofs" in physics which are wrong. If their foretellings and experimental results gainsay one another, then the assumptions are wrong. However, here are two famous examples of how proof "failure" and physics interact:
Mathematics through all these processes helps us clarify the fine minutiae of meanings present in our sometimes taken for granted physical assumptions.
Lastly of course, Mathematical Proof can be seen as a kind of "investment adviser": it tells us where to put our hardest work and other resources in experiment. Unless you have reason to finely question a physical assumption A that seems already backed up by experiment, an experiment that tests the logical outcomes of combining assumption A and assumption B by mathematical reasoning is a much better use of time and work than a foretold result which can be shown to be logically equivalent to the already experimentally supported assumption A.
Footnote: I foresee automated proof development and checking by things like software proof systems as important to physics in the future. As I understand it, many parts of String Theory suffer from this kind of problem, that they are hard to review by one or few reviewers alone. Fortunately although proof development systems are themselves astoundingly complex, the proof checking software itself is a simple parser that can be reduced to one or two pages of code and thus can be thoroughly experimentally debugged, so it doesn't matter how proofs are constructed as long as they are deemed valid by the parser.