It is difficult to provide a straightforward answer to the question in the title. However, things have changed with respect to “packing peanuts”. These are Styrofoam packing materials. Traditionally they were made of polystyrene which is made from petroleum-based substances. However, my research tells me that since the early 1990s starch-based packing peanuts have been developed as a more environmentally friendly alternative.
Therefore nowadays the actual material itself appears to be non-toxic to cats and to people. But I would certainly check this point because I don’t know whether all Styrofoam packing peanuts are made from starch. There may be some in the system which are petroleum-based and these I would argue would be toxic to a cat or to a person if ingested.
There’s an interesting story written by Barry Scheps on Quora.com (he used to write for my website some time ago). About 20 years ago he discovered that his cat was munching through the white “packing-particles” or packing-peanuts that had been left in a box on the floor for days. He saw that there was no adverse effect so we ate one himself! He decided that they were foodgrade starch-based packing peanuts and quite tasty!
That said, I don’t think it’s a good idea if your cat munches through packing peanuts made from crop-based materials as opposed to chemicals. One problem is that perhaps your cat won’t masticate them enough. Perhaps he may swallow them whole or semi-whole. And if he ate too many there may be a blockage. What I’m saying is, and what I’m concerned about, is that there would appear to be a potential for a blockage to be created which would be serious. And in any case packing-peanuts are not meant to be cat food!
Therefore, the conclusion is that modern packing peanuts are not, on the face of it, unsafe for a domestic cat but I wouldn’t recommend cats eating them. There is a side issue about polystyrene. Polystyrene is used today to make insulating boxes (cooling boxes) for various products. I recently wrote about a cat rescue organisation using polystyrene boxes as homes for feral cats. It’s a good idea but I see a very faint potential for cats to eat bits of polystyrene that may fall off or be scratched off these boxes. This polystyrene would be chemical-based and although they may pass through a cat in their faeces there is potential for health issues in my opinion.