Polyethylene (PE or PEX)
Polyethylene (PEX) is probably the least harmful option. It will leach small amounts of VOCs for some years. Different brands cause different odors and leach different chemicals - this has not been thoroughly researched yet. Precautions you can take are to avoid drinking warm water from the tap or filter your drinking water with RO/carbon on distillation.
When running lines to a tiny house use PEX instead of a garden hose or RV hose. RV hoses are made of PVC.
PEX tubing is also the tubing of choice for radiant floor heating. Radiant Floors are a great heating option, as ducts can pose problems for those sensitive to mould and dust. I prefer electric underfloor heating as I have seen PEX floor heating spring a leak.
With PEX make sure to select the right diameter for your application.
Copper can be considered for those extremely sensitive to plastics. However, there are also health risks associated with it. Those with EMF sensitivity are concerned when copper pipes pick up stray electric currents.
Copper can be filtered from drinking water with a reverse osmosis system.
For people very concerned about mould they sometimes run their water lines inside the house. In this case you would use copper.
Copper is much more expensive than PEX and more prone to bursting if it freezes.
PVC or ABS (Outgoing Pipes)
Either one can be used on outgoing water pipes. Your codes may require one or the other.
Make sure that when glue is used at junctures in the pipes it is done outdoors or while the house can be totally aired out. Make sure your contractors are aware of the toxicity of the glue and to be extra careful with spills and clean-up. The glues cure very quickly, however. So they should be tolerable very soon.
Where plumbing meets the wall, it should be sealed with non-toxic caulking.
For more information see Prescriptions for a Healthy House: A Practical Guide for Architects, Builders & Homeowners