When your overflow pipe is leaking, it may indicate that you've got a problem with the float valve.
The float valve is usually a plastic ball at the end of a metal arm in the tank that's usually found in the loft. It controls the water level of the tank and will cause your overflow pipe to leak if it's faulty.
The most common problems include a leak in the float, which means it needs replacing, or a fault with its washers, which will need refitting.
If you do have a leak you may wish to contact an approved plumber.