Conductivity of a semiconductor at high temperatures
A number of different things are going on in this scenario.

First, the carrier concentration does increase steeply, although for Si at its melt temperature it is still only about 10$^{20}$/$cm^{3}$, much lower than a metal. 

In addition, the band gap is decreasing with temperature (not from carriers, but from changes to the lattice spacing). From a graph in Sze's Physics of Semiconductor Physics, Si goes from 1.12eV at room temperature to a little under 0.9eV at 900K. Extrapolation (probably a bad idea) means that somewhere above 2700K there will be little to no band gap. However, this assumes the physics extrapolates linearly, which it doesn't. (Well, liquid silicon is metallic. However, there is still a large increase in conductivity at the melt point).

Finally, the phonon population increases with temperature, which will increase scattering, decreasing the conductivity (just like a metal). This can be seen from a graph of carrier mobility (also in Sze) with temperature. The mobility drops roughly as T$^{-2.2}$ or so. 

Getting the balance of all of these just right under extreme extrapolation, well, that is hard. It is actually easier to point to papers on the electrical characteristics of undercooled liquid silicon (since that has been measured), than to superheated solid silicon (which has not been observed). - for ev