The history of mathematics over the last 200 years has many occasions when the fundamental assumptions of an area have been shown to be flawed, or even wrong. Yet I cannot think of any examples where, as the result the mathematics itself had to be thrown out. Old results might need a new assumption or two. Certainly the rewritten assumptions often allow wonderful new results, but have we actually lost anything?
Note I would like to rule out the case where an area has been rendered unimportant by the development of different techniques. In that case the results still hold, but are no longer as interesting.
I wrote up a longer version of this question with a look at a little of the history: http://maxwelldemon.com/2012/05/09/have-we-ever-lost-mathematics/
Edit in response to comments
My thinking was about results that have been undermined from below. @J.J Green's example in the comments of Italian algebraic geometry seems like the best example I have seen. The trisection and individually wrong results do not seem to grow into areas, but certainly I would find interesting any example where a flawed result had built a small industry before it was found to be wrong. I am fascinated by mathematics that has been overlooked and rediscovered (ancient and modern) but that is perhaps a different question.