long story short, transparent plastic containers tend to imbue their contents with a chemical called bisphenol-a (aka bpa), which is a compound similar to estrogen which may impact human physiology in a wide array of ways[0], or chemicals that are structurally related that aren't well studied but could be similarly damaging[1]. this applies to all transparent plastics and isn't specific to aeropresses

[0] https://en.wikipedia.org/wiki/Bisphenol_A#Health_effects

[1] http://www.motherjones.com/environment/2014/03/tritan-certic...

Leaching chemicals. Here's an article I found after a quick google search, http://myplasticfreelife.com/2011/04/bpa-free-does-not-mean-...

It doesn't mention heat, but I presume leaching increases with heat.

I think it's because it is possible for materials in certain plastics to break down in the presence of heat.