Are there certain types of foams I should not be cutting? There are so many types of foam and of as many different chemical compositions. Not that this should be taken as standard but the wiki for hot wire cutter has this for the first sentence:


  A hot-wire foam cutter is a tool used to cut polystyrene foam and similar materials.


Emphasis mine

Standard pink crafting foams appear to be made from polystyrene as well. Polystyrene is a really broad plastic and it is used in lots of things though. Standard expanded packing foam (the white bubble stuff) is made from polystyrene. So I should be able to work with any of those? 

I keep foam packing materials in case I need to return products. I need a way to identify foams that are safe to use a hot-wire cutter on. I think a smaller list to limit myself to certain foams as supposed to listing which ones I should avoid (assuming there are any?)

Keeping to the actual foam variants: Is that what I should limit myself to? Polystyrene? Some fumes can more toxic than others. Some foams might be more flammable than others.
Yes there are certainly foams you should exercise caution around when cutting. And an easy way to check what cautions you need to take when working with different foams is to look up the foam's specific MSDS report.

MSDS stands for Material Safety Data Sheet.

"A Material Safety Data Sheet (MSDS) is a document that contains information on the potential hazards (health, fire, reactivity and environmental) and how to work safely with the chemical product. It is an essential starting point for the development of a complete health and safety program."

Definition taken from the Canadian Centre For Occupational Health and Safety

https://www.ccohs.ca/oshanswers/legisl/

Here's a couple of examples for MSDS reports for different "foams". The first is the MSDS for rigid foam insulation the other is for Polystyrene Foam. The last link is for the rigid pink foam insulation.

Some foams are safe to cut (knife, saw or heated wire) the other foams require personal protective equipment (PPE) just to be around.

Learn the material you want to work with and study its MSDS sheet before doing anything with it.

https://www.kasteel.ca/wp-content/uploads/2015/07/MSDS-Polarfoam-PF-7300-0-Finished-Foam.pdf

http://www.styreneproducts.com/media/files/SPI%20EPS%20MSDS(1).pdf

http://insulation.owenscorning.com/assets/0/428/429/434/896b50d3-24ef-4263-997a-62927ac03237.pdf

I think there was another question here on the exchange which talks about the importance of MSDS sheets. If someone knows the link then please edit this answer to include that post.