The valve seat inside a tap can become worn in time. If, after replacing the washer on a tap, it continues to drip, this may be the problem. Limescale and other debris can scratch and damage the surface. Even with a new washer, a proper seal will not be made.
There are two options:
Regrinding a valve seat is a simple job and is carried out using a reseating tool. These are readily available from DIY stores and Plumbers Merchants.
Switch off the water supplies to the tap. Open both to allow the excess water to drain off then put the plug in the plug hole so that small parts don't disappear down the waste. Remove the tap head by removing the cover if there is one, and releasing the retaining screw. Tap designs vary and the process for removing the tap head may be different.
Use a spanner to undo the nut on the headgear - remember to hold the tap body firmly whilst doing this. Remove the headgear from the tap body and place to one side.
Screw the reseating tool into the opening and adjust the cutter so that it is in contact with the valve seat. The surface can the be smoothed by turning the tool handle.
Valve seating set
Alternatively, a valve seating and jumper set can be purchased. The new plastic valve seating is placed into the existing seat. Replace the jumper and washer with the new one. Reassemble the headgear in the tap body and retighetn the retaining nut. Replace the tap cover.
Now close and open the tap a few times. The closing action will force the new seat liner into the original valve seat. These kits may take a day or two to 'bed in' fully so don't be overly concerned if there are a couiple of drips still.
Switching on the water supply
When you've finished regrinding the valve seat or fitted the jumper and insert kit and the tap has been reassemmbled, switch the water supplies back on. Test each tap allowing the water to flush through