Positive Grid current in valves
What actually happens when you exceed the manufacturers grid current rating?

Generally, the grid melts.  Finer grid wires are an advantage, because they have lower capacitance and because they intercept fewer electrons on their way to the plate.  So if a tube is designed to operate at low or zero grid currents, the grid will be made of finer wire, and no particular effort will be made to pull heat out of the wire.