Sadly, the answer is "no". Hot water, both in the boiler and in the radiators, is very corrosive IF there is oxygen in it. Water from your well or town has dissolved air, including oxygen (it's why fish survive in water). Once the water is in the system, the oxygen combines with the boiler/radiator linings, corroding it.
There isn't a lot of oxygen in any one volume of water, and the oxygen in the system will soon get used up, stopping the corrosion. However, if you keep changing the system's water (typically through a leak, or with your suggestion wholesale by taking the system water for your washing machine) then you're continually replenishing the water and thus its contained oxygen. Bingo: corrosion city.
There's almost certainly other good reasons as well (e.g. the water in a boiler is fairly low-pressure, so you wouldn't get great water flow; the boiler fill valve isn't meant to run constantly, etc.), but having to replace your boiler and radiators every few years would trump it all.