TITLE: Why the electrons below the Fermi level do not conduct electricity?
QUESTION [16 upvotes]: Physically, why is it that the electrons need to excited above the Fermi level to conduct electricity? In other words, why is the current zero when the electrons lie below the Fermi level? Does Pauli exclusion principle play any role here?

REPLY [1 votes]: "Physically, why is it that the electrons need to excited above the Fermi level to conduct electricity?"
They don't.  This is a popular misconception; the premise of your question is mostly wrong, as are all 6 answers supplied to date (June 22, 2018)!
A simple proof that the other answers are wrong is to consider that the number of electrons above the Fermi surface is zero at absolute zero.  Therefore, if they are correct, and I am incorrect, the resistance at absolute zero would be maximum.  For most substances, the resistance is actually minimal at absolute zero, as can be seen experimentally in this graph:

In a normal and pure metal, the electrons below the Fermi level are nearly superconducting, and move without any resistance to speak of.
Conceptually, one can view the superconductive fraction of the electron cloud like hockey pucks in a game of air hockey, with the disturbances from ions being small compared to the sizes of the pucks, allowing a smeared or average force of zero to it:

However, electrons above the Fermi the level have a relatively high energy and an effective size small enough to be able to "see" (and therefore interact with) the periodic ionic field due to the screened, tightly-spaced ions.  
In my air hockey example, these high-energy contributions toward the electron cloud are small enough to get pushed backwards a little before going over an air spout or sliding without an air layer.  This interaction produces momentum exchange, which, because the electrons have to be thought of as a cloud distribution, not to mention that splitting of the cloud would cause a strong field, gives the entire electron cloud a net drag force measurable as resistance, thus slowing it down to the drift velocity and preventing it from net acceleration.
In a non-pure conductor, impurities cause warps in the lattice, which the electron cloud will view as drag.  
In my air hockey example, this is like dirt in the surface which pucks rub against.
In a nonconductor, all of the electrons are trapped on the atoms, and there is no conduction except for special cases like high fields (e.g., lightning) or external radiation (e.g., the photoelectric effect).  
In my air hockey example, the number of free hockey pucks is zero, as they are all stuck between the air holes.
In a semiconductor, the number of free electrons in the conduction band increases with temperature, as electrons are forced out of the $0$K Fermi sphere in momentum space.  In this case, raising the temperature can increase the conductivity if the conduction band was relatively empty.
In my air hockey analogy, the number of free pucks is small but increases as one increases the temperature, dislodging them from their ions.  This happens for atoms with intermediate valances, like carbon, germanium, silicon, etc.
For this case, the premise of your question would seem to make sense, except that your question uses the word "need" without any restriction instead of something like "..need in semiconductors..." Therefore, it is wrong, along with all the other 6 answers to date for failing to notice this as well as for saying incorrect things, like that conducting electrons always need to jump to available states above the Fermi level (the current #1-voted answer), the momentum of electrons below the Fermi level fully cancel each other out (the current #2-voted answer), etc.
Incidentally, once a conduction band in a semiconductor becomes populated, further increases in temperature should decrease conductivity and increase resistance as phonons can interact more strongly with the free electron cloud.  
In my air hockey example, the free, conducting pucks become small enough to have to slide without an air layer. Therefore, even in semiconductors more electrons being above the Fermi level does not always correlate to lower resistances or more current flow.
Turning now to your question about the relevance of the Pauli Exclusion Principle, it comes into play because electrons forming the cloud are trapped like the proverbial particle in a box of size determined by $Z$ and the ionic spacing.  If these electrons are unpaired, they act like Fermions.  This controls the number of electrons that can fill the lowest levels, which in turn controls the number of electrons that can fill the upper levels, like the conduction bands in semiconductors.  Without this, there wouldn't be electrons up so high in these conduction-band energy levels, and nature would be much different.
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