.+ve grid current is the grid getting hit by too many electrons. it commences at a little less than 0vg wrt cathode. this is the current which occurs as you drive the output tubes too hard, and the grid swings above the cathode potential. the electrons flowing in the plate current are no longer repulsed by the -ve potential, instead being attracted by the +ve potential.
this current occurs in perfectly operational valves. if anything, an amp which is not capable of driving into + grid current will actually end up biasing the output tubes COLDER, clamping them off.
this is in contrast to -ve grid current, which is another beast entirely.
this happens when there is excess gas in the tube. the flow of plate current at electron voltages above 25v or so will impact any gas molecules with enough energy to dislodge another electron. the gas mollecule has now turned into a positive ion, which will be attracted to the nearest NEGATIVE electrode, which is–you guessed it–g1.
when the +ve ion eventually collides with the control grid, it pulls an electron back out of the grid circuit. this makes the grid a bit more +ve than it just was… and THIS is what leads to the really sticky issue:
the control grid’s more +ve potential will result in an increase of plate current. more plate current = more ionization = more -ve grid current = more plate current, etc etc. the tube runs away, and will destroy itself in short order if plate power is not removed.
ideally the initial evacuation of the envelope was effective, and the getters are doing their job. in this case the current is TINY. all valves have an imperfect vacuum so they all have SOME degree of -ve g1 current, but it is an ideal to have none.
this ionization current is what leads to the maximum grid circuit resistance specification given in the tube data sheet. more -ve grid current = less max permissible g1 circuit resistance. therefore with a valve prone to runaway the g1 dcr should be as low as possible (i.e. tranny coupling, cathode follower drive).