Milk Cartons and Ice Cream Containers are made from paperboard coated with a waterproof plastic, generally polyethylene (plastic). There are drink cartons lined with "renewable and plant-based" resin, however the resin behaves like conventional plastic and does not break down in municipal composting systems. Renewable materials are generally understood to be made from matter that can be regenerated or regrown in a human lifetime vs non-renewable conventional plastics made from petroleum. Ice cream container lids with thick plastic around the edges belong in the garbage.
The Ecology Center has written more about milk cartons here.