It's actually kinda bad enough if they start breaking down plastics. We use plastics in a lot of places specifically because they're resistant to microbial breakdown. This includes electrical insulation, medical equipment, food containers, and so forth. A world where plastics start to rot at the pace of wood is a world in which we have big problems.
Unfortunately, the way we've gone about things it seems like an inevitability that a successful plastic-eating microbe would evolve. Plastics are goddamn everywhere, and they're chock full of yummy hydrocarbons. Any organism that specializes in this food source has very good prospects for thriving.
Plastics are a terrible food source for microbes. The two most common plastics polyethylene and polypropylene are just chains of C2H4 and C3H6 respectively. No nitrogen, phosphorus, sulfur, potassium, or any other elements necessary for life.
In a world full of carbon dioxide in the air and photosynthetic life forms, having access to yet another carbon and energy source isn’t much of an advantage except in extreme environments like hydrothermal vents.
EDIT: Also seawater: https://en.wikipedia.org/wiki/Plastic_degradation_by_marine_...