This claim is most likely related to the presence of certain controversial chemicals present in some kinds of plastic. The most discussed is certainly Bisphenol A (BPA).
The claim about microvaving in plastic bottles is probably related to the fact that BPA leeches faster out of plastic with hot water than with cold water. One study found that
Exposure to boiling water (100 °C) increased the rate of BPA migration by up to 55-fold
Bisphenol A can mimic estrogen and thereby cause a variety of health effects.
In the review "Environmental causes of cancer: endocrine disruptors as carcinogens" the authors state the following about the occurence of Bisphenol A:
The xenoestrogen BPA is one of the EDCs that has been most thoroughly studied. BPA is found in various consumer products including baby bottles, reusable water bottles and reusable food containers, polyvinyl chloride stretch films, papers, cardboards and in the epoxy resins lining the insides of food cans.
On the carcinogenic properties of BPA they state:
Rats exposed prenatally to environmentally relevant doses of BPA show an increased number of intraductal hyperplasias (precancerous lesions) that appear during adulthood, while high doses induce the development of carcinomas in situ.
There is still some conflict about the interpretation of the studies. The Endocrine Society is concerned, but many regulatory bodies like the EFSA and FDA consider the current allowed levels safe. Although recently the EU banned BPA from baby bottles.
