I think it would be very unlikely to work. Weeping willows, like most willows, send down extensive root systems that try to find water. I doubt that they will grow very well if they can't access moist soil.
We planted a weeping willow cutting five years ago beside our pond which has a bank about four feet above water level. For the first four years it hardly grew, but then the roots got down to the water and it really took off.
The real question is where are you growing your weeping willow? It will only root towards the drain if it has no other alternative. I certainly wouldn't grow it near a building as they make rampant growth and take a lot of moisture from the soil.
To quote:
"Weeping Willows (roots and all) can grow extremely fast if given the proper nutrients and conditions for growth. They can grow up to several feet per year and will develop a spread equal to their height in many instances. They need to be spaced at least 40 ft apart. Their roots are highly invasive and will most definitely affect nearby water and sewer lines. The roots can exceed the growth of the expanse of the tree's branches by 3 to 4 times, so even planted a good distance away the roots are likely to eventually head toward your septic bed and clog it, unless you have a giant lot. "
http://en.allexperts.com/q/Trees-739/Planting-Weeping-Willow-Trees.htm