Why would your car be dripping water after running the heater?

Most heater settings on a modern car with Air Conditioning will cause the A/C compressor to kick on. This is to remove moisture from the inside air and prevent frosting of the windshield and Windows. The air conditioner coils accumulate the moisture which drips down and through a small tube or hose and out the bottom of the car. That's what you are seeing, if it's clear water.

On the other hand, if it's coolant (antifreeze) you have a leaking heater core.