oven repair, is it possible the gas line is clogged?
Cleaning is always a good idea after a mishap like this, but if it does not immediately fix the problem, you should call in a professional repair service. A fire in an appliance, especially a gas fed appliance, needs a pro to ensure that the unit is still safe to operate.

Many plastics can form a pretty solid mess when melted onto metal. It may be difficult or prohibitive in time to clean some of those parts. Also, the burning plastic may have damaged a thermostat or electrical connects. Whether anything needs to be replaced is again a decision that should be made by a professional specifically familiar with this unit and who can directly inspect the damage (or lack of damage).