Neutral object and electric field

Do you mean the object as a whole is neutral, or the charge density is zero everywhere? Because if it's the former, it may experience a force if the field isn't uniform (eg, if the field is strong near the positive charge and weak near the negative), but if it's the latter there will be no force.

 

If conductor YES else NO.

 

An insulator with a nonuniform distribution of charge, such that the net charge is zero, can experience a torque (or "couple") due to an applied uniform electric field, but of course that is not the same as "a net force." But what if the applied field in not uniform? Imagine an insulator in the form of a long rod. At one end, it has charge +Q, and at the other end it has charge -Q. In between, it is uncharged. So it is neutral overall. Now turn on an electric field that is perpendicular to the rod. If the field is uniform, the rod will start spinning, but its center of mass will remain fixed. But suppose the field at the +Q end has strength E and the field at the -Q end has strength 2E. The negative end of the rod will experience a force twice that of the other end, so not only will the rod start spinning, but its center of mass will in fact accelerate, so that it does in fact experience a nonzero net force.