Unlikely. Plastics are composed of large bio-compatible molecules. The main danger of plastics by themselves is displacement--bits of plastic getting lodged in tissue or cells and sticking around because of that bio-compatibility. That's a real concern, but hardly on the same level as arsenic which chemically disrupts the function of our cells.

There are secondary effects of plastics which accumulate and later release other toxins, but it's a bit disingenuous to blame that on the plastic.