My laundry is in the basement, and my thermostat is upstairs. It's always cooler downstairs than upstairs, so I don't think the wet laundry causes the furnace to run more often.
My drying racks are directly under the furnace vents, and clothes dry astonishingly fast. I dry outside in the summer. The only iffy months are April and October, when it's cool and wet outside, and not really so cold that the furnace runs much inside.