However, I had the oven on, and I failed to notice this exhaust pipe, which is connected to the oven, under the burner. The hot gasses from the oven melted the plastic to form the mess in the first photograph.
I felt foolish, but when I told my landlord about it, he suggested a simple solution. I followed his advice to heat up the plastic-coated burner again and use a grill brush to scrub off the melted plastic. It worked well, as you can see in the photograph above, and the burner is back on duty.