There are environmental issues associated with the sourcing of either: fossil oil is toxic, but e.g. monoculture tree plantations come with environmental issues as well (fire-prone, insect issues, not a good habitat for other species, and of course felling is problematic too).
Paper is biodegradable quite easily, plastics usually aren’t. Even plastics sold as biodegradable usually need certain conditions to break down. Paper can properly be recycled, tiny bits of plastics can usually only be burned.
Plastics can be made lighter than paper, i.e. you need less material. Some manufacturers put PFOA coatings on paper used for fast food, despite PFOA never breaking down under normal circumstances.
The best thing fast food companies could do is to stop creating drinks and drinks containers that work best with a straw and just save straws entirely.