Compass froze or overshot actual magnetic bearing - NO.
Instability of indicators - NO.
Digital VOR navigation bearing display errors up to 5 degrees - NO.
VOR navigation To/From indicator reversal - NO.
VOR and ILS course deviation indicator errors with and without a failure flag - NO.
Reduced sensitivity of the ILS Localiser receiver - NO.
Background noise on audio outputs - YES, although nothing major.
That's purely a response from me in regard to my own aeroplane. I have considerable disdain for bad science, and have tried my level best to prove that any of these assertions hold any water by deliberating dialling my home line to maximise RF activity then shoving the phone as close to these "sensitive" instruments as possible. Nothing... absolutely nothing.
That said, as mentioned above I'm not a scientist and I'll keep them switched off during serious stuff, like flying instrument approaches. They are however very useful on a nice VMC day for calling a taxi from 20nm out.