What is a food-safe level of exposure of plastic containers to heat?
You ask:


  I'd like to know if the information passed around the internet regarding the release of BPAs into the contents of plastic containers is grossly over-exaggerated or if there is a grain of truth to it that I should be concerned about to a degree.


In the United States, the Food and Drug Administration (FDA) has published  Bisphenol A (BPA): Use in Food Contact Application, in which they state:


  FDA’s current perspective, based on its most recent safety assessment, is that BPA is safe at the current levels occurring in foods. Based on FDA’s ongoing safety review of scientific evidence, the available information continues to support the safety of BPA for the currently approved uses in food containers and packaging.


From the link (above), you can go to the 2014 Updated safety assessment of Bisphenol A (BPA) for use in food contact applications, where they conclude that the "no observed adverse effect level (NOAEL)" for BPA is 5000 $\mu\mathrm{g}$ per kg of body weight per day ($\mu\mathrm{g}$/kg-bw/day) and that exposure estimates are "0.2 $\mu\mathrm{g}$/kg-bw/day (mean) and 0.5
$\mu\mathrm{g}$/kg-bw/day (90$^{\rm th}$ percentile) for the adult US population aged 2 years and older." The estimated dietary intake (EDI) is given as 1.1 $\mu\mathrm{g}$/kg-bw/day for population under 2 years old and 0.5 $\mu\mathrm{g}$/kg-bw/day for population greater than 2 years old. Thus, the exposure levels are on the order of a thousand times less than the NOAEL.

The European Food Safety Authority (EFSA) has recently (January, 2015) revised the tolerable daily intake (TDI) of bisphenol-A (BPA) from 50 $\mu\mathrm{g}$/kg-bw/day to 4 $\mu\mathrm{g}$/kg-bw/day. See this press release and this FAQ for more details.

The summary from the press release is:


  EFSA’s comprehensive re-evaluation of bisphenol A (BPA) exposure and toxicity concludes that BPA poses no health risk to consumers of any age group (including unborn children, infants and adolescents) at current exposure levels. Exposure from the diet or from a combination of sources (diet, dust, cosmetics and thermal paper) is considerably under the safe level (the “tolerable daily intake” or TDI).


The scientific opinion on the risks to public health related to the presence of bisphenol A (BPA) in foodstuffs, used by the EFSA to bolster the down-revision of the TDI for BPA, has been summarized in a very nice form suitable for the lay public, available here.

What about heating?

In one publication (Assessment of Bisphenol A Released from Reusable Plastic, Aluminium and Stainless Steel Water Bottles. J. E. Cooper, E. L. Kendig, and S. M. Belcher. Chemosphere 85(6), 943-947 (2011)), the authors state:


  At room temperature the concentration of BPA migrating from polycarbonate bottles ranged from 0.2–0.3 mg/L. Under identical conditions BPA migration from aluminium bottles lined with epoxy-based resins was variable depending on manufacturer ranging from 0.08 to 1.9 mg/L. Boiling water significantly increased migration of BPA from the epoxy lined bottles. No detectable BPA contamination was observed in water stored in bottles made from Tritan™ copolyester plastic, uncoated stainless steel, or aluminium lined with EcoCare™. The results from this study demonstrate that when used according to manufactures’ recommendations reusable water bottles constructed from “BPA-free” alternative materials are suitable for consumption of beverages free of BPA contamination.


and they conclude


  As shown previously by our results and those of many others, BPA migrates into water stored in polycarbonate plastic, especially when heated to high temperatures. Relatively high levels of BPA are found to be released into drinking water from reusable aluminium bottles with an epoxy-based liner. The amount of BPA released from the epoxy coating was greatly increased by heating. No detectable BPA was observed in water to migrate from unlined stainless steel, EcoCare™ copolyester lined aluminium, or copolyester Tritan™ plastic water bottles. To avoid exposures to BPA unlined stainless steel, copolyester lined aluminium or copolyester plastic drinking bottles should be used.


So, should you be worried? In my opinion, that's largely a function of whether or not you use plastic that contains BPA. A few years ago, it might have been difficult to find BPA-free plastics, but there is no shortage of BPA-free items available now. It's easy enough these days to remove any doubt from the equation by simply using containers that don't contain BPA.