Can a conductor become an insulator?

Sometimes. In most materials, no. However, things like vacuum tubes and MOSFETs in electrical amplifying equipment will change between insulators and conductors based on a "control" voltage. Computers rely on this behavior for their most basic principles of operation.

Technically, all insulators are conductors because they conduct electricity through them. The difference is that insulators conduct a lot less electricity. Thus, for a conductor to become an insulator, we would have to raise the electrical resistance to a level that we could consider "insulating". In most materials this can only be achieved by changing the temperature, and the temperature change required for this is impractical to a great degree. Almost all electrical devices, however, rely on the ability of electrical switches to change from insulator to conductor to insulator again at will. Depending on the voltage applied to the "control" part of a MOSFET transistor, its resistance could be extremely high (several thousand or million ohms) or extremely low (about 0.08 ohms). Also the voltage will have a great deal to do with it. At a high enough voltage some material may brake down and conduct a large amount of current, llike a zener diode.