Question

When is the net torque acting on an object zero, and yet the net force is nonzero?

Answer 1

This happens more frequently than the converse. For instance pushing a block across a table.

Net torque changes the rotational velocity of an object. Net force changes the velocity of an object.

Pushing moves the block horizontally and the block has no rotation assuming low friction. Sufficient friction between the table and block would produce a net torque which would cause the block to flip.