Nah. Our curtains are hanging over our radiators too. They are not that hot, even on full blown, to actually light anything on fire. It “burns” your fingers, as in you can not touch them, but not in a real, burn wound kinda way.
We regulaly put things on them to dry or e.g. orange peel to have a nice smell in the house.
Edit: if you are not familiar with them: they have hot water circulating in them which is warmed by a central unit somewhere in the house. Nothing electrical there.