My furniture also hides the length of most of my drapes, but there are 2 places where they're visible and anything other than floor-length would have looked silly. I went ahead and hung them anyway and figured I could always hem (ok, safety pin...I rarely sew!) them up when the heat came on. We could stand having it look weird for a few months out of the year.

It turned out to not really be a problem. They get warm, but not hot, and they cover just a tiny part of the baseboard heater. They are a very light burlap-ish fabric, though; I might not have tired it with something heavy and rich like velour.

Well, radiators are under windows because cold air coming off the windows falls, and warm air rises off the radiators. You would have a cold convection air current in the room if they were locates somewhere else. ( I work in HVAC)

Curtains help insulate against the window edge, which can block air leaking in during the winter. They offer shading in warm weather to lower your solar gain and save on cooling costs.

You should be fine generally to have curtains at your windows with radiators. Most hot water systems are 120-140 degree water. If you have space heaters then you need to be very careful.

I re-read what I posted above, and want to be more clear.

To clarify, if you have space heaters, infa-red, or other heat types that warn against contact with flammables, you need to NOT have curtains or furniture on or near. In this situation also be mindful of tipping, blowing, and falling hazards that could place flammables next to the heat source.