This may sound unusual, but it is actually a rather common problem. All of your bathroom fixtures are connected to the same supply pipes inside the walls of your bathroom. The toilet, tub, shower, and sink are all connected to the line in the same way, so a problem with one faucet may be related to the pipes leading to one of the other bathroom fixtures.
The most common issue that would cause a faucet to leak as the toilet is flushed is high water pressure. Normal residential water pressure is between 50 and 70 psi. If the pressure exceeds 80 psi, it can cause complications. Under normal circumstances, the water flows in from the supply line and refills the tank whenever you flush the toilet. Once the tank is full, the pipe between the water supply line and the tank is automatically closed to keep the toilet from overflowing. When this happens, the water is forced backward. If you have high water pressure, water can be forced through a worn or deteriorated seat washer, which can cause the faucet to drip. The drip will normally stop once the pressure decreases; however, it can cause more problems over time.
If your home is equipped with a pressure-reducing valve, you may want to have a professional inspect it to see why it is not properly regulating your water pressure.