Does a red hot metal reflect light in the same way as a cold metal?
The light that you see from hot metals is incandescence.
Incandescence is caused by high temperature. The high temperature promotes electrons to higher energy levels. These electrons then relax and emit light.

On the other hand, the "light" (shiny) that you see from room temperature metals  is caused by re-emission/reflection of other light. Light from another source is absorbed by the metal and re-emitted, or the light from another source is reflected from the surface of a metal.

The reason as to why metals look shiny have to with plasmons. If I recall correctly, most materials will look "shiny" at a certain wavelength range. It just happens that metals are shiny in the visible wavelength range.

What determines what type of light a material will reflect is determined by its plasma/plasmon frequency. Temperature does seem affect the plasmon/plasma frequency. Several studies confirm this.

In general, it appears that the plasmon/plasma frequency decreases with increasing temperature. In other words, the material is able to reflect a narrower range of light as the temperature is increased. This would mean that if you had a material that is shiny in the visible light range, you could make it become dull by heating it. Of course, this is a simplification that doesn't take into account other factors.

However, I wasn't able to find an equation that directly relates temperature and plasma frequency. How the plasma frequency changes with temperature seems to depend on the material and the size of the material (nanoscale, bulk, etc). Furthermore, there are ab initio studies that suggest that the opposite must happen: the plasmon frequency increases with temperature.