Will energy in an LC circuit become 0 if its is disconnected when the capacitor is fully discharged?

If switch is opened when capacitor is fully charged energy of LC system remains same.

If it's a series LC circuit, then the energy will be $E=frac{CV^2}{2}$ all residing in the electric field of the capacitor. If it is a parallel LC circuit of an ideal inductor and capacitor, the energy will remain the same but it will be exchanged between the electric field of the capacitor and magnetic field of the inductor at the resonant frequency.

If switch is opened when capacitor is fully discharged energy of LC system becomes 0.

The only place where energy, if any, can be stored is in the magnetic field of the inductor, $E=frac{LI^2}{2}$ and that would require current in the circuit prior to the switch opening. Once the switch is opened, current cannot flow. The energy stored in the inductor will be dissipated in an arc between the switch contacts as they open, caused by the high voltage induced in the inductor in response to the attempt to disrupt the current by the switch.

Hope this helps.