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.
Another example from the same area is an upper estimate of the number of these limit cycles for quadratic systems. The wrong 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 many people believed that the Riemann Hypothesis was proved by Stiletjes, who published an announcement. Stieltjes died at a young age.