Food grade polyethylene tubing is unlikely to leach anything so long as it is not in the process of being destroyed by ultra-violet light. You can, however, leach copper into water. Tiny amounts may be good for humans, but there is a limit that I haven't bothered to look up, so anyone concerned should do so.

Residential reverse osmosis systems, widely used to remove chemicals in water, use PE tubing and fittings of various NSF approved plastics such as polypropylene. The RO membrane filters are themselves contained in plastic housings.

I would be more concerned about the source of the bottled water. Does it come from a source where leaf decay can occur in the water with the result of a vast array of possible organic chemicals? Is plastic piping used in the bottling plant (as noted above) or in the plant's well pipe? Distillation from glass retort to delivered bottle might be 'pure' so long as there weren't any chemicals in the water that had a lower vapor point than water. It is really hard to make water ultrapure. In any case, bottled water typically keeps or adds some minerals for taste.