I'd remove the glass from the door and clean it with an appropriate solvent.

Thanks jon1270. I tried using mineral spirits on a small area of the frosted part and it left the glass kind of smeared looking. I was afraid to go any further without some professional help. Is frosted glass as impervious to chemicals like unfrosted glass? Can you use a razor blade to remove tape residue or paint overspray etc?
Thanks in advance.

Not all frosted glass is the same. Some is sand blasted or acid etched. This type can be cleaned with MEK, Lacquer thiner, Acetone, etc. followed by any window cleaner without any damage. The other type of frosting is sprayed on. A translucent white paint. Cleaning this type with solvents will remove it.

Mineral spirits will not work after the stain is dry. You will need one of the stronger solvents.