What is causing toilet supply water leaks above the coupling nut?
The washer under the nut is probably leaking and the easiest/best solution's would to be replace the pipe. 

Why

The reason it only leaks when you have the supply on is that there is no significant back pressure on the fittings from the tank as the tank has an air gap and the filler is at the top of the tank not submerged at the bottom ( which would cause back pressure)

P.S. DON'T use mutigrips, use a spanner or shifting spanner, on a plastic nut (or even brass) you can deform the nut by the crushing force of the mutigrips as you tighten the nut.