Was there a case in history when a theorem was incorrectly proved, but people still used it without realizing that theorem was wrong? Although I am not mathematician, but I would guess that there are numerous theorems which have very complicated proofs, and I would further assume that a lot of them have real world applications (e.g physics, economics etc)
Since even the smartest do make mistakes, I start wondering whether there was a least one case in history when someone came up with a conjecture, ostensibly provided correct proof (but actually proof was wrong), then people started using this theorem, and later someone had realized that theorem, is in fact, wrong.
Was there such a case?
 A: There are finitely many such cases. A detailed list all such well known cases can be found in this link.
One example: The following enjoyed the status of a theorem for 15 years before the error was detected and the statement of the theorem was revised.
Grunwald (1933), gave an incorrect proof of the erroneous statement that an element in a number field is an $n$-th power if it is an $n$-th power locally almost everywhere. George Whaples (1942) gave another incorrect proof of this incorrect statement. 
However Wang (1948) discovered the following counter-example: 16 is a $p$-adic 8th power for all odd primes $p$, but is not a rational or 2-adic 8th power. In his doctoral thesis Wang (1950) gave and proved the correct formulation of Grunwald's assertion, by describing the rare cases when it fails. This result is what is now known as the Grunwald–Wang theorem. 
A: Since we want an answer where an incorrect result was extensively used before it was detected and corrected, I am quoting an example from applied mathematics/physics. Hence posting a as a separate answer. 
Issac Newton discovered the laws of gravity and motion and his theory and equation worked very well for precise calculation of the motion of heavenly bodies. But there was one exception; the planet Mercury always deviated from the calculation predicted by Newton's theory of gravity and for 250 years astronomers still used wrong calculations because even though Newton's equations of gravity were not very accurate for Mercury, there was no alternative set of equations to describe the universe in a more accurate way.
At the start of the 20th century, Einstein discovered that nearby a massive body such as he sun, the laws of gravity deviated from Newton's equations due to the bending of space and time by the strong gravitational field of the massive body. Newton's equation would only work when we are far from a massive body but fail for a planet like Mercury which is right next to the sun. Einstein's general relativity corrected Newton's equation and the modified equations perfectly described the motion of Mercury. In effect, the equation of general relativity was a correction applied to Newton's equations.
