What happens when net torque is zero?

[jayadds]

I know that net torque is the sum of torques in a given situation. So take this as an example: If I used a wrench to turn a nut counterclockwise and it succesfully moves, is the net torque going to be non-zero? If so, then can someone please give me an example where the net torque can be zero.

Furthermore, if net torque is zero, does that mean the object doesn't rotate?

 

[haruspex]

It's no different in this regard from the linear case.

A nonzero net torque will result in an angular acceleration.
When the nut starts to rotate, from stationary, it is being accelerated, so there is a net torque. But very soon it's rotating at a steady rate, so the net torque is back to zero. The torque you continue to apply is equal to the dynamic friction.

In the period during which you were building up the torque, but the nut was still not yielding, your applied torque was exactly balanced by static friction.