There are at least two Hilbert problems that were considered to be solved, but the proofs turned out to be incomplete, as pointed out by Yulii Ilyashenko.
In 1923 Dulac published a 140+ page memoir purporting to show that a polynomial vector field on the plane has only finitely many limit cycles, the second part of the 16th Hilbert problem. The memoir was difficult to read, but the claim was generally accepted until in 1981 Ilyashenko found a serious gap. Full proofs were obtained independently by Écalle and Ilyashenko around 1991. Read the full story.
Existence of linear differential equations having a prescribed monodromic group was the subject of the 21st Hilbert problem, also known as the Riemann-Hilbert problem. From Wikipedia article:
Josip Plemelj published a solution in 1908. This work was for a long time accepted as a definitive solution; there was work of G. D. Birkhoff in 1913 also, but the whole area, including work of Ludwig Schlesinger on isomonodromic deformations that would much later be revived in connection with soliton theory, went out of fashion. Plemelj produced a 1964 monograph Problems in the Sense of Riemann and Klein, (Pure and Applied Mathematics, no. 16, Interscience Publishers, New York) summing up his work. A few years later the Soviet mathematician Yuliy S. Il'yashenko and others started raising doubts about Plemelj's work. In fact, Plemelj correctly proves that any monodromy group can be realised by a regular linear system which is Fuchsian at all but one of the singular points. Plemelj's claim that the system can be made Fuchsian at the last point as well is wrong. (Il'yashenko has shown that if one of the monodromy operators is diagonalizable, then Plemelj's claim is true.)
Indeed in 1989 Soviet mathematician Andrey A. Bolibrukh (1950–2003) found a counterexample to Plemelj's statement. This is commonly viewed as providing a counterexample to the precise question Hilbert had in mind; Bolibrukh showed that for a given pole configuration certain monodromy groups can be realised by regular, but not by Fuchsian systems.