Nope. The only way for the tank to overflow is for the flotation device — the ballcock — to be removed from the tank, causing the valve to never close.
You seem to be thinking of only one force when in fact there are at least two forces acting on the ballcock. If you were to empty the tank instantaneously by cutting out the bottom of the tank, what would happen to the ballcock? It would fall on its own.
The ballcock has an upward force (buoyancy) acting on it, causing it to float when the tank is not empty. It also has a downward force (gravity) acting on it, causing it to fall, when the tank is empty.
So the reason why the ballcock causes the valve to open and close is due to a pressure differential.
Wouldn't that cause the tank to overflow instead?