I would like to point out that a falsifiable pseudoscience would eventually be corrected because someone would publish an experiment which falsified it. I am of the mind that large-scale evolution is such an unfalsifiable pseudoscience. But here I am interested in the larger question. I only bring up that particular belief because I am sure that, if I did not, someone would use it in an ad hominem attack.
To the direct point, I think that such an error could remain uncorrected indefinitely. There is no perfect test to distinguish between science and pseudoscience. In fact, the determination is made by existing scientists. Going against the grain (unless you can prove it) will get you ignored, dismissed as a crackpot, shunned, or something similar. But one cannot prove that no possible observation will falsify an idea. Believers in an idea will insist that the idea is potentially falsifiable. Look at the followers of any pseudoscience. They will state that they have conducted tests and that differing results would have "falsified the theory." And they are sincere. They really believe it. But the significance of data are only determined after the fact. Negative data are dismissed as unimportant or explained away. Positive data are heralded as important confirmation. The followers of a pseudoscience genuinely cannot see that they have set it up to be unfalsifiable. So, if a pseudoscience was ever accepted, how could it be detected? Once it was in, how could it be rooted out?