One very famous error was made by H. Dulac in his 1923 paper which was supposed to prove
that a polynomial system of differential equations in the plane has finitely many isolated
limit cycles. (This was one of the Hilbert problems). It was believed until 1980-s that
he proved the result. 

The error was the following. He proved that certain function, say analytic on $(0,1]$
has an asymptotic expansion of the form
$$f(x)\sim \sum_{n=0}^\infty a_n x^n,$$
in the usual sense that asymptotic expansions are understood.

He concluded that $f$ has finitely many zeros on $(0,1]$.

The mistake was found in the early 80-s and corrected. The new proofs
of the finiteness theorem for limit cycles, which are accepted as
correct are very long and difficult. But there are at least two different published proofs,
one that I know is by Ecalle, and another by Ilyashenko.

A related problem is to estimate the number of isolated limit cycles in terms of
the degree of the equation. This is still unsolved. And there were published wrong solutions.