Why does attaching the case to the PCB ground plane cause the audio to cut out on my cassette player?
Your PCB is attached to a metal plate.  There are components on the PCB between the PCB and the metal plate.
Very likely there is a component that is just a little too tall.  As long as the screw isn't completely tight, there's a tiny gap between the component and the plate.  If you tighten the screw really tight, the component touches the plate and causes a short to ground.
Take out the PCB, turn it over, and look for parts that stand up higher than the others.  See if those &quot;too tall&quot; parts can touch the metal plate.