Does a well-lit mirror weigh more than an unlit mirror? If you weighed a mirror in a room with no light, and then weighed a mirror in a well lit room so that the mirror reflects light, would the weight be different?
 A: What about heating up the mirror? I assume not all light is reflected but also absorbed. Energy is related to mass.
Here is a summary of real experiments that show how kinetic energy in an object contributes to its gravitational mass:
arxiv:gr-qc/9909014
A: Photons are elementary particles and they do have an invariant mass, when there are more than one of them: the addition of their four vectors. Example: pi0 into two gamma, those two gammas have the invariant mass of the pio0.
A dark mirror in principle should show a smaller mass  because the photons which hit and reflect , at the time of hitting have an invariant mass which is overlaid on the mass of the mirror. 
This is separate from momentum conservation showing up on a weight scale, radiation pressure. And separate from the absorption increase of mass .
A: Yes, if the mirror was on the floor and facing upward. The transfer of momentum as photons recoil from it and are absorbed by the surroundings would make it heavier.
A: Where is the light source?  If the light source is above the midpoint of the mirror the mirror will weigh slightly more.  However, if the light it below the mirror it's weight is going to drop.  (This effect goes away if the mirror is absolutely perfect.  Real world mirrors will absorb some photons, though.)
There will be a considerably stronger (but still infinitesimal) sideways force imposed on the mirror by the light bouncing off it.  Even if the room is evenly lit this still pushes on the shiny side vs the non-reflecting back side.
Note that both of these factors apply relative to the mirror's orientation.  Dirk Bruere's answer is force #2 applied to a mirror laying down.
A: While some light energy may be absorbed by the mirror, the only light that has an effect on the image is the light that bounces back off.
For the same reason a ball touching a bat isn't considered as an increase in bat weight, the mirror's weight is not considered as an increase when the light hits the glass.
Further, the light photons that are absorbed (and are not affecting the image appearance) affect nearly every object in similar ways. Thus it is not special to the image either.
