Ignore CitationLover, he knows not of which he speaks.
The real answer: It depends. It depends where the error comes from. It could come from the Omni-Bearing Selector (OBS), It could come from the VOR receiver, or it could come from the CDI.
If the error is due to the OBS, it will not affect a localizer as the OBS has no effect on localizer display.
If the error is due to the VOR receiver, it will have no effect on the localizer information as the Localizer is a seperate receiver which works on a completely different principle then the VOR receiver. Note, even though the ILS receiver is almost always housed within the NAV radio and shares a frequency selector, it is in fact a different receiver.
If (and only if) the error is due to a miscalibration of the CDI, then the error will affect the CDI when a localizer is tuned. I would say that a miscalibration of the CDI is probably the *least* likely source of a one dot error.
regards