Tank stuff was pretty much covered. The result was the only unrealistic thing about it. There's basically three outcomes.
Breech closed - Grenade at best damages gun, but unlikely since grenades don't have that much blast power. The fragments would have to be removed before firing the next shot however, else that would certainly damage the gun - but probably not the point of rendering it useless, mind you. it would just become horribly inaccurate, which isn't really a big deal since these people can't aim for shit in the first place.
breech open - Either grenade rolls inside the tank and explodes. Shrapnel takes out crew. Nothing else happens, tank just stops moving.
Or grenade causes a chain reaction by igniting the stored ammo inside the turret. The whole turret will explode in a giant fireball, as seen in the video Leijia posted.
The only thing that will certainly not happen is a small flame starting to burn out of the hatch.
tl,dr: grenades can disable tanks through the barrel.
The Governor seems to be capable of turning people into bloodthirsty psychopaths. But then again, this is Alabama, right? How many people who live there now are violent, hostile gun nuts? If it were Walking Dead in Canada, we'd be looking at a different show.
Of the m60 Patton, more than 15k vehicles have been built. That means throughout its service history of nearly fourty years, hundreds of thousands of people were trained on it. It's not that unlikely that in a random group of 50 adult people, one of them may have received M60 Patton training.