How can I trace the path of plumbing lines inside walls without cutting holes?
Metal detector? Thermal imaging? There are utility locators that pick up a signal conducted through metal by a transmitter. Even with non-metallic pipe sometimes you can remove the faucet and push a wire down the pipe. Acoustic detection via electronic means is also a method used.

There are contractors the offer location services that have sophisticated equipment. Their fees are proportionate to their equipment costs... high. It's likely much cheaper to cut and patch holes.