Question
can a weeping cherry be planted near a house?

Answer
You will want to remember that this tree will end up with a circumference of anywhere between 20 ?40 feet.  So plan accordingly. When planting trees of any kind, it is important to allow them ample room for root development (roots will extend about half again as far as the canopy of the mature tree). It's also important to give them adequate spacing for good air circulation, which is important in preventing disease. Tree roots can damage a foundation, so again it is wiser to plant the trees out in the open at a safe distance from both the foundation and any underground utilities. At a minimum, determine the mature width of the tree, and plant the tree at least 3/4 that distance from the house.
Most small ornamentals like weeping Cherry do not have an agressive root system, so damage to a foundation would probably not happen. I would recommend at least 8-10' from the house.