[Edited to correct the Galileo story] An old example of a plausible result that was overthrown by rigor is the 17th-century example of the hanging chain. Galileo thought once said (by inspectionthough he later said otherwise), I suppose) and Girard claimed to have proved, that the shape was a parabola, but . But this was disproved by Huygens (then aged 17) by a more rigorous analysis. Some decades later, the exact equation for the catenary was found by the Bernoullis, Leibniz, and Huygens.
In the 20th century, some people thought it plausible that the shape of the cable of a suspension bridge is also a catenary. Indeed, I once saw this claim in a very popular engineering mathematics text. But a rigorous argument shows (with the sensible simplifying assumption that the weight of the cable is negligible compared with the weight of the road) that the shape is in fact a parabola.