I think you have that wrong. High screen current in a pentode happens when the plate voltage gets much lower than the screen voltage. This is caused by having too high a load impedance.
If you tried to bias yourself into an idle state with Vg=0, the tube would conduct heavily and die from too much plate dissipation, unless you lowered the screen voltage greatly. In any case, since the tube draws grid current when Vg > 0, and the usual guitar amp driver circuits can't supply grid current, it woud hardly work.
For triodes, the failure would presumably happen due to overdissipation of the plate, since there's no screen to burn out.