That's a great question, especially for those of us in colder climates! The short answer is: it depends heavily on the material. Most standard plastic lids, especially cheaper or older ones made from brittle plastics, can become a real problem. In deep freezes, they can become stiff, difficult to open, and are prone to cracking or snapping if you force them. The plasticizers that keep the material flexible can lose effectiveness over time, making them more vulnerable.
Metal lids, like those on some commercial bins, generally fare better against the cold itself but can face other issues. Hinges may freeze shut, and if there's any moisture, ice can seal the lid tight. Corrosion can also be a factor over time.
The best performers are usually high-density polyethylene (HDPE) or other "all-weather" rated plastics designed for temperature extremes. They retain more flexibility. My top tip for winter is to give the lid's seal and hinges a light spray with a silicone-based lubricant before the deep cold sets in. It prevents ice from bonding to the surface. Also, try to keep the bin in a sheltered spot if possible, and avoid letting wet, heavy waste freeze solid right under the lid, which can make it impossible to open. A little preventative care goes a long way in keeping your bins functional through the winter months.