There are pretty much inert because they're ignored by most proteins and are largely stable enought to resist chemical damage (oxydation is the big one). This is the reason we make so much of them, they're useful as containers and it's cheaper to make new containers rather than reuse the old ones.

That being said, it could happen that some biological processes start using plastic as an energy source and/or process it for parts. This is already done in labs with plastic eating bacteria today, where the bacteria produces enzymes capable of breaking the molecular bonds in some plastics.

The main factor is that there is considerably less plastic readily available than there is carbon dioxide, because the carbon dioxide is just there, but in a hypothetical future where the earth is covered in several inches of plastic, chances are that single cell organisms that thrive off of plastic could realistically evolve.