Is this known to work? I would have thought that running a whole tube at high temperature would cause the grid to emit thermal electrons, creating a current between grid and plate. Even though the grid doesn't have the low-work-function coating that the cathode does, its surface area can be larger and it's usually more negative than the cathode so I'd expect substantial currents.

If this does work, you could save a lot of power by building heated, insulated enclosures for entire tube appliances rather than heating each cathode and cooling every tube.