if it is a white plastic pipe, and it only drips when the boiler if firing, and the boiler is quite a modern one (5 years old, say)
then:
it will be a condensing boiler (the hot fumes coming out of the flue outside will be steamy in cold weather, and no hotter than a cup of tea)
and the white plastic pipe is almost certainly the condensate pipe, that takes condensation (from the steam generated by burning the gas) which forms inside the boiler chamber, and takes it to a convenient drain.
This condensate is slightly acidic (about the same as tomato juice) so if you collect a bit in an eggcup, and rub it on your taps, you will probably find it cleans the limescale off. However it is a bit polluted with Products of Combusition, and it should not be leaking in the house (under certain unlikely circumstances, with a combination of faults, flue gases might leak through that pipe).
So you ought to report it as a boiler fault. If you have a rented house the landlord should fix it, if not you can engage a boiler engineer or a skilled plumber (this part is just a fancy drainpipe and is not within the gas part of the boiler, so you are not obliged to have the leak repaired by a gas-safe engineer). If you have a maintenance contract they will fix it. It is not an emergency.
The white plastic pipe probably has solvent-cement joints in it, these are very difficult to take apart, most likely the repairmain will cut out the damaged section and replace it with a new bit. A boiler engineer might open up the boiler casing and replace the pipe all the way from the vessel inside, to the drain, which would be neater. People who are not gas-safe registered are not supposed to open the case.
BTW - it is very commendable that you have the condensate running (by the sound of it) to a convenient nearby waste pipe in the bathroom. Many people have these pipes running outside the house where they freeze up in very cold weather, causing the boiler to flood or switch off, which is very inconvenient in cold weather. Many installers say their customers wouldn't want the extra cost and inconvenience of having an internal waste pipe put in; I say they will once they've had an external one freeze.