As long as you make sure that the wood is isolated from the source of moisture as much as possible (or better still remove the source) then any decent quality wood properly primed and painted should be OK.
I would also consider getting the wood treated first.
If we're talking a cast iron radiator, it's a good process.
First you need to remove the old finish. You can get it sandblasted, or you can get in there with some sandpaper for a few hours. A wire wheel on an angle grinder can get out a good bit, or you might have luck with chemicals to strip them. If you do use chemicals, you'll have to wait for them to totally dry, then sand the bare metal of the radiator just to make 100% sure all of the residue is gone and the surface is rough enough.
Next, you'll want to clean with a deglosser, to make sure all the sanding residue is gone, along with any other junk in there. The paint won't bond so well without it, and with something that expands and contracts as much as a heater.. you want to make sure it's thoroughly clean.
Now it's time to protect your walls and floors. Get a drop cloth, cut a slit in it and slide it over the unit. Tape it to the walls, tape it to the floor, and tape up as much of the slit you made as you can.
You'll then need to prime it, and you'll have to chose the primer based on the paint you end up with.
Time for paint. Oil based paint should hold up, but you'll probably have better luck with spray enamel paint. Use many thin coats rather than few thick coats. Make sure to read the specs on whatever paint you decide on, make sure it can withstand up to 220 degrees - Just to be safe.
Alternatively, maybe you can look into building a radiator cover.
Best Answer
None. Why would you paint over good paint?
Otherwise, one. Recent experience with the likes of Behr Premium Plus and other modern products has left me very impressed with coverage and durability.
Of course, this requires good technique. You can't ever "stretch" the paint by pushing it out further than you should. A loaded roller covers something like 10 square feet at most on a smooth surface.