What would cause the round plastic sight hole to the furnace burners to melt around the edges.

Probably overheat.
If not melted through just a bit of heat mark on the edges it could be normal with age.

BUT it could also indicate improper combustion air control or needs a professional adjustment.

2 much heat

Blocked flue vent, not enough combustion air, bad safety sensor, improper gas/air mixture, all of the above. Get a technician to inspect it.

A natural gas furnace (NAT) operating on propane (LP) without the proper orfice conversion.