There are many accounts of tanks firing multiple times, to ensure their target is dead. For instance:

As we were moving, I fired once. Then we stopped and I fired two more shells to make sure they wouldn't fire at our side. All three of our shells penetrated, one under the gun shield and two on the side. The two side hits went completely through and out the other side.[1]

.

Suddenly the peace was shattered by Gray Boyce’s voice over the air saying vehicles had been heard from the road, and next moment a Mark IV danced into view. Hell for leather, actions automatic, load, fire, reload. Hit it with probably third shot and pumped in a few more until it brewed up. [2]

There are what are known as "functional" kills, where the vehicle is effectively disabled and unable to fight, and a true kill where it is truly dead. A functional kill might be hitting the treads and wrecking them - the tank might still be able to turn and elevate the gun, but it can no longer move and is essentially an exposed pill box.

Generally speaking, there is usually a good visual indication of either a function or true kill in the way of visible damage, fire, explosion, etc. There are many stories of US tanks hitting German tanks and visually seeing the rounds bounce off - they would know to keep firing!

Likewise, the more lightly armored Sherman tanks were known for cooking off when hit - the germans would know to stop firing.

In between such obvious visual cues, there are a variety of tell-tale signs. Internal fires, even if not large enough to set off the ammunition, will ooze smoke or fire from hatches and vision slits. Low-order explosions may pop hatches open, which helps point one. Guns and turrets may be at odd angles.

Or the tanker may do the safe thing and put another round in just to be safe.