Effect of temperature on the Fermi level in a semiconductor

While the Fermi level is dependent on the temperature, most easily seem through the relation to effective band density of states, something else more drastic will happen if you raise the temp. Regardless of what your doping levels might be, the number of intrinsic carrier pairs will start increasing. As the number of intrinsic carriers surpass the doping carrier concentrations, the Fermi level will be pulled back to midgap. What you can do, however, is dope beyond a bands effective density of states, I.e. degenerate doping. Then your Fermi level will rise above the conduction band, or drop below the valence band. And yup, these sort of heavily doped semiconductors behave somewhat like poor metals. Other neat things like Esaki diodes can use this principle.

In an intrinsic semiconductor (compared to daFireman's answer related to doped semiconductors), the direction of Fermi-level shift due to increased temperature depends on electron and hole effective masses (related to the effective density of states).

http://britneyspears.ac/physics/basics/basics.htm Check eq. 26.

Suppose we keep Fermi-level constant, and increase temperature. If the density of states of valence band is larger, than for conduction band, there will be more holes than electrons and the system would be charged (impossible). Therefore, the Fermi-level needs to shift up, closer to conduction band, to compensate for this.

So in principle, the Fermi-level can go over Ec or Ev, one just needs enough temperature. In practice, system might melt before that.