Is it really true that valence band is completely filled at zero temperature? Is it really true that valence band is completely filled at null temperature?
Indeed, I would think that if we apply an electric field, this would give some energy to the electrons from the valence band, so would they be prevented to leave the valence band to go to the conduction band, thanks to the energy from the electric field?
I don't see why would the electrons know if the energy that they receive is due to temperature or from electric field source?
 A: If a perfect semiconductor/insulator is at zero temperature, than indeed all its electrons are in the valence band, and exciting them to the conduction band requires, as a minimum, the gap energy, $E_g$.
If a uniform electric field is applied, then, obviously, there will be degeneracy between the valence band and the conduction band states, separated by distance of order
$$eEd=E_g$$
(More precisely, we have to recalculate the states of the crystal, taking into account the electric field, since the translation symmetry is broken now in one direction.) This degeneracy results in electrons tunneling from the valence band to the conduction band, which is a kind of dielectric breakdown, known as Zener effect.
However, in practice this effect is observed only at very large electric fields or in artificially created materials with small band gap (superlattices). The reason for that is that the tunneling is possible only when the tunneling length is comparable to the coherence lengths, which is usually rather short in  real materials.
Note also, that in a real material the imperfections (dislocations, impurities, etc.) will typically result in both valence and conduction band having tails extending well into the gap, so there might be always a residual concentration of electrons and holes, even at zero temperature - but too low to result in detectable electric conductance.
A: The temperature we're talking about here isn't the temperature of the room where you do the experiment, or even the temperature of the atoms in the material. It's the temperature of the ensemble of electrons in the material.
If you had a sample of semiconductor with its electrons at 0 K, and you did something to it (apply an electric field or anything else) that promotes some electrons to the conduction band, then we'd say the electrons are no longer at 0 K.
It's possible they're not in a (quasi) thermal equilibrium state and we can't even describe them with a temperature. Or that their energy distribution is close enough to what we expect in some equilibrium state and we say they have a temperature $T$ with $T>0$.
The important thing is that temperature is a simplified description of the distribution of energy in some material or ensemble of particles. So if you change the energy distribution (by promoting some particles to a different energy level), you've changed their temperature (or possibly made it impossible to define their temperature).
