The FDR is dependent upon the same sources as the flight crew. So if the air data system is disabled then the same inaccurate data will be recorded by the FDR.
The only circumstances where the FDR can record data different to the crew is when the source data is ok, but the display system is faulty. For an extreme example, if all the cockpit displays were unservicable, the FDR would record the correct info, but the crew would be unable to see any of the data.
Even if the FDR is recording the same (unreliable) data that is of value because
(a) it indicates that the soruce of the data was the problem; and
(b) with the luxury of time and a calm ground environment, it can be possible to deduce reality even from multiple sets of misleading data - because we know the aircraft conforms to basic physical laws, you can cross-compare data and synthesise, say, a flight path, then check against other data. The hypothesis which best fits the known facts can then be assumed valid, and thus "right" can be sorted from "wrong".