The dial lights are festoon lamps and will short if touching ground. If you are not using festoon lamps, that could be the cause of a short. Some people mistakenly use fuse lamps. They are usually a little longer and the bases are wider than festoons, expanding the mounting clips farther. Both aspects place live contacts closer to possible grounds.
The reflective paper should not be conductive. It is not intended to electrically insulate but it does serve to reflect the light toward the sides of the dial glass. That paper will always be burned. This was definitely a design shortcoming on these units. Sony did not account for the cumulative heat from the dial lamps (or perhaps they had no idea their receivers would last for 5 decades!)
The reflectors are easily replaced. They can be reproduced using what's left of the old reflector as a pattern and cutting / folding replacements from non-conductive, reflective, card stock (readily found at a craft store.)
The incandescent festoon lamps can be replaced with LED festoon lamps. The LEDs put out much less heat, so they will not burn the paper reflectors. Also, if the LEDs are on just one side of the lamp (as many are), the lighting will be very directional, and can be directed toward the edge of the dial glass.
If your reflectors are burned, the green tint on the edge of the dial glass is probably faded or gone, as well. It is easily restored using glass paint, from either an art or craft store.
Properly restored, your dial should look something like this:
