Negative impact of wrong or non-rigorous proofs The recent talks of Voevodsky (for example, http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf), which describe subtle errors in proofs by him as well as others, as well as the famous essay by Jaffe and Quinn (http://www.ams.org/journals/bull/1993-29-01/S0273-0979-1993-00413-0/) and responses to it (http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00503-8/), raises for me the following question:
What are some explicit examples of wrong or non-riogour proofs that did damage to mathematics or some significant part of it? Famous examples of non-rigorous proofs include Newton's development of calculus and the latter stages of the Italian school in algebraic geometry. Although these caused a lot of dismay and consternation, my impression is that they also inspired a lot of new work. Is it wrong for me to view it this way?
In particular, I'm told that people proved false theorems using Newton's approach to calculus. What are some examples of this and what damage did they do?
 A: First, three possible areas of damage (though there are surely more): 


*

*Subsequent results that make use of these "proofs" (especially when the claim is not true);

*Using the methods of the incorrect proofs, when, in fact, this is where the problem lies;  and 

*Causing others to lose trust in the institution of mathematics (e.g., questioning rigor more broadly). 
Second, an explicit example: Du-Hwang's proof of the Gilbert-Pollak Conjecture, which was later shown to contain a serious gap. The go-to for a "proof" of it was a text by Ivanov and Tužilin, but since the error in the proof has been discovered, those two have gone on to explain not only where the Du-Hwang proof went wrong, but also why attempts to patch it up have been unsuccessful. To this latter end, see their arXiv note here from February 2014.
For a related MO post, see here (where I believe the top comment is from Ivanov) and a link to the note mentioned above (which contains references for further reading).
More generally, one might reasonably expect that realizing a proof is wrong took some insight, and where there is insight, it seems quite possible that there will be an inspiration for new work. Whether or not that work will lead to newfound success is sure to occur on a case-by-case basis; I'm not sure that the error in the Du-Hwang proof has led to anything of great import at this time, though it has renewed a bit of interest in the area of Steiner minimal trees.
A: Since the question is specifically about damage:
I think that what really causes damage to a mathematical area is when an important result is claimed by someone prominent in the field, but the proof is never completely written.
Younger researchers are then likely to spend a lot of time and energy "cleaning up the mess", for little credit.
Things are even worse when there is some freedom of interpretation of what might have been proven.
A younger researcher might want to use the announced result for some other purpose, but they might use a version of the theorem that ends up not being the one that got proved.
When a proof is (widely) accepted to be wrong or non-rigorous, or when someone retracts the claim of having proved a given result, that's when things are getting better for a field.
A: A proof being wrong can mean many things:

By 1932, when the Hungarian-American mathematician John von Neumann claimed to have proven that the probabilistic wave equation in quantum mechanics could have no “hidden variables” (that is, missing components, such as de Broglie’s particle with its well-defined trajectory), pilot-wave theory was so poorly regarded that most physicists believed von Neumann’s proof without even reading a translation.
More than 30 years would pass before von Neumann’s proof was shown to be false, but by then the damage was done. The physicist David Bohm resurrected pilot-wave theory in a modified form in 1952, with Einstein’s encouragement, and made clear that it did work, but it never caught on. (The theory is also known as de Broglie-Bohm theory, or Bohmian mechanics.)

The problems is the interpretation of what has actually been proved. I don't like the No Free Lunch Theorems for Optimization, because their assumptions are unrealistic and useless in practice, but the theorem itself certainly feels true (but in a less trivial way than what is actually proved). And the conclusion is deeply flawed. It claims that there is no difference between a buggy implementation of a flawed heuristic and a correct implementation of a reasonable solution strategy. The conclusion should rather be that we should explicitly specify what our solution strategy is supposed to achieve, not just claim that it is a good black box search strategy.
A: There are several examples of wrong proofs which were believed to be correct for some time, but I would not say that they "did damage to mathematics".
One of the most famous examples is Dulac's proof that a 2 times 2 polynomial system of
differential equations in the plane has finitely many limit cycles. A gap was found 60 years later, and after some substantial efforts the proof was fixed. Now we have two different published proofs, both are quite complicated. The story is told in great detail 
in several publications of Ilyashenko. His book
MR1133882  contains a complete proof as well as the history.
Another example from the same area is an upper estimate of the number of these limit cycles for quadratic systems. An incorrect proof was published by Landis and Petrovski, but soon retracted. The problem is not solved to this day, to the best of my knowledge.
There are many other examples. In the beginning of 20-s century some people believed that the Riemann Hypothesis was proved by Stiletjes, who published an announcement.
Stieltjes died at a young age, and never published his proof.
If some one in really interested in the result, s/he would make all efforts to understand the proof, and eventually the things will be sorted out. If no one is seriously interested, there is no damage to mathematics anyway:-) Remember, huge efforts were made in 19 century to make Calculus rigorous. Many Fourier arguments were also doubtful. Best mathematicians of 19 and 20 century made efforts to put Fourier analysis on a rigorous basis.
A: It has to be said that in the history of mathematics sometimes quite new profound ideas  suddenly arise, so that tools, methods, and foundations are still lacking in a first phase of the new theory. This is the case of certain parts of Analysis  at    Cauchy's times,  and it is also the case of the "Italian Geometry", which later grew into  modern Algebraic Geometry.  In such special occasions, I wouldn't say that non-rigorous proofs have a negative impact on the theory; on the contrary, they bring  the attention of the mathematical community on it, and turn into a call for well-founded methods.
A: 
In particular, I'm told that people proved false theorems using Newton's approach to calculus.

I'm intrigued by this claim, but it seems very vague, both because there is no information about what the theorems might be and because it's not clear how widely accepted these false results are claimed to have been.
First off, I don't think it makes much sense to consider Newton in isolation from Leibniz. Newton didn't fully develop the calculus. His results were scattered in a variety of places and are not complete or systematic.
The rigor and logical validity of Newton and Leibniz's calculus were vigorously debated from very early on. Archbishop Berkeley published The Analyst in 1734, seven years after Newton's death. He claimed that although Newton's results were correct, they were derived through incorrect methods: "I have no Controversy about your Conclusions, but only about your Logic and Method." This would seem to suggest that if otherwise competent people arrived at incorrect and widely known results based on the Newton-Leibniz methods, it didn't happen during Newton's generation or the generation after that, since clearly a rabid critic like Berkeley would have made a big deal out of such a thing.
Blaszczyk at al. have an interesting revisionist take on the early history of the calculus. In their reading:

Leibniz’s heuristic law of continuity was implemented mathematically as Los’s theorem and later as the transfer principle over the hyperreals ..., while Leibniz’s heuristic law of homogeneity... was implemented mathematically as the standard part function ...

If you buy this account, then Newton-Leibniz calculus had a fully formed, well-defined, and consistent set of methods that are isomorphic to some subset of the methods of NSA. For example, say we calculate, in the Leibniz style,
$$d(x^2)/dx=[(x+dx)^2-x^2]/dx=(2x\,dx+dx^2)/dx=2x+dx{}_{\ulcorner\!\urcorner}2x.$$
The symbol ${}_{\ulcorner\!\urcorner}$ is Leibniz's notation for what he called "adequality," which Blaszczyk argues is the same as NSA's standard-part relation; today we'd write the final step as $\operatorname{st}(2x+dx)=2x$. This is a completely valid calculation if interpreted in terms of NSA. Of course the notation  ${}_{\ulcorner\!\urcorner}$ never caught on, and practitioners of the calculus traditionally just wrote $=$, which made the practice of discarding squares of infinitesimals seem logically suspect. But that doesn't mean that the methods were wrong, just that they were traditionally written in a way that may have obscured their correctness.
Blaszczyk, Katz, and Sherry, "Ten Misconceptions from the History of Analysis and Their Debunking," 2012, http://arxiv.org/abs/1202.4153
