If the paper becomes overly coated with plastic or wax, you should replace it with new paper until all of the mess is absorbed into paper and removed from the other surface. The paper and plastic or wax will be very hot! When you are finished, make sure to let them cool in a safe place before you dispose of them. Also - be sure to test a small, or not as easily noticeable, area of the fabric or carpet first - you don't want to melt them.
Before you get the iron, depending on the size of the melted plastic, you could actually just go at it with a hairdryer - on its highest setting and melt it off, and use a butter to scrape it out. After that, use oven cleaner spray. Spray it on the melted patch, leave it to set and then try gently tugging/dragging it out of the carpet hairs. This might actually work better on smaller melted patches, but the only downside is that you may end up with a bald patch on your carpet! Nail varnish remover is also a good one - but this one only has maximum effect if you apply it straight away before the melted plastic has time to set. The properties of acetone allow it to break down the plastic making it easier to clean.