Yes, it can. It can clog their gills, making their feeding less efficient, or interfere with their reproduction through chemicals that leach from the plastic particles. Source: I study bivalves
Oysters actually need clean/unpolluted water. What they eat and clean is sediment, phytonutrients, and phytoplankton. Well at least that’s what I was tood by an oyster farmer. They clear the water which is good for the sea grass, and the small fish, which is good for the big fish and so on.
They are confirmed to be able to gather nano-plastics as larvae, and microplastic as adults, though no studies have looked at nanoscale particles in adult oysters from what I could find. They may ingest the plastic or package it up in mucus to expel as "pseudofeces", trapping the plastic particles in the sediment. Source: I study bivalves
It's generally seen as pretty hard to maintain filter-feeding bivalves long-term in an aquarium tank, in terms of diet (diverse phytoplankton), pH/alkalinity, and substrate (many are pretty specific in terms of conditions in which they burrow). You might have better luck with Corbicula sp. ("Asian clams") which are quite generalist in their food, even being able to deposit feed by scraping the substrate. They are however quite invasive, so please don't release them into the wild! Oysters on the other hand are usually found in mostly brackish or fully marine conditions, and would likely not be able to thrive long-term in fully fresh water. Usually they can only survive short bouts of freshening. Source: I study clams