Good points all around. I've been air-drying in a prewar Astoria steam-heat one-bedroom for years, and there are two separate problems usually getting mixed up here.
First, direct contact with the radiator. Gowanus_G is right: the paint, dust, and sometimes old metal smell transfers to fabric when it's draped right on top. It can also leave rust marks or yellow lines on lighter stuff if the radiator has ever wept. Use a rack that sits at least 6 inches away, or a flat mesh screen on top if you have a ledge-style radiator. Don't use the cheap wooden rack directly over the radiator—wood can hold paint/dust and sometimes resins get warm.
Second, speed. EastHarlem_E's wet dog smell is almost always moisture hanging around too long before the fibers fully dry. Wool and heavy cotton blends are the worst. For those, I do a second high spin in the washer, then shake them out hard before hanging. Spread them out, don't fold them over the rungs. Point a small fan across the rack, not directly at it, and open the top and bottom windows a crack if you have a double-hung. The air needs somewhere to exit.
Beth, your old pennies smell with a hoodie might be a combo of too much detergent and direct heat. Try less detergent, a full extra rinse if possible, and don't hang the hoodie over the metal. If it already smells, a wash with a cup of vinegar in the fabric softener slot should knock it out. Keep an eye on the radiator top before heating season: wipe it down once with a damp cloth after it cools.
This setup won't make your place feel like a laundromat, but it'll keep most stuff from smelling like the building's basement. The dehumidifier is the last resort, not the first.