How do I remove polyethylene that's melted to a nylon carpet?
Using a household iron works, with caveats.

First: several sheets of paper towel between the iron and the carpet. These are here primarily to protect your iron (and your shirts!). I was hoping that it might also might be something that the plastic preferred to stick to, but that didn't happen.

Second: no steam, and set the heat on wool (about 1/2). I slowly worked my way up the heat dial: the synthetics settings weren't hot enough (as expected), the cotton setting (3/4) wasn't appreciably better, and the linen setting (full) made the plastic adhere more tightly to the carpet (fortunately, I tried this near the end, to get the last bits out, and it actually had the benefit of making them disappear).

The process: put the paper towels over the spot, then let the iron sit for 20-30 seconds. This will soften the plastic, such that it can be removed either by pinching between your fingers or using tweezers. Fingers are preferred, as tweezers tend to tear out small pieces. In retrospect, a dull knife (such as a table knife) would have been a good tool: it would go on bottom, pulling the plastic away, with a finger on top to apply pressure.