Lithium conducts electricity in its solid and molten states. Lithium exhibits metallic bonding and as such its valence electrons are delocalised in the liquid and solid state allowing electricty to be conducted.
Lithium oxide only conducts electricty in its molten state. It is an ionic compound and in solid Li2O the ions are fixed in position in an ionic lattice, the ions are not mobile and therefore cannot conduct electricty. However, when molten the ionic bonds are broken and the ions are mobile allowing electricty to be conducted.