My friend published a paper. After few months he found some error present. He asked the journal editor for a correction, made the changes and replaced the old version with the new one. The old journal paper is still present in websites other than the journal. Will these websites replace the old with the corrected new paper? If not, will it be a problem?
It depends on the third party website. If they are constantly monitoring the main journal, it might happen, but I would assume that this is not true for every website. On the other hand, in most cases the journal should only be available via the journal web site thanks to the copyright regulations of that journal. And people citing the work should always look for the original source.
Therefore I would say it is not a problem of your friend, maybe it can be a problem for people working with unreliable sources. But I would not care.
This is why journals don't like to make corrections to published papers. The journal can fix the version on their website, but once electronic versions of the paper have been sent out, they can't be recalled, and it's beyond the journal's control to replace all of them.
As OBu said, it's up to the third party websites to replace the old paper with the new one, and this depends too much on the website to predict. As for whether it'll be a problem, I'd guess no: if the error is major enough that it'll be a problem, the journal should absolutely not change the paper but should issue a corrigendum. Since that didn't happen my guess is that the error is too minor to worry about.