Do coiled power cords heat up and cause fires?

Amperage is the issue, not voltage, and how many and tight the coils. In essence coiling cable creates an electromaget via amperage, which induces heat, through resistance. just like an electric motor. Thats why motors have cooling fans.
I have seen spooled welding cables melt, catch fire. However, constant current welding cables are used at 17-36v and amperages 20a to hundreds.
Relax. if there was an issue with a coiled 120-240v cord the NEC has dealt with it.