The answer depends upon the plants themselves.
Herbaceous perennials – Plants which have frozen all the way to the ground, and are not woody, should be cut back to roughly two inches tall. I don’t recommend cutting them flat with the ground, as it’s easy to forget where your plants were, come spring! Herbaceous perennials whose tops have died, but have new leaves still healthy near the ground, should be pruned back to that point. Many perennials put out new leaves in the fall that stay up all winter and should be left alone.
Woody perennials – Woody perennials which are hardy should be cut back halfway at this point, such as salvias, Turk’s cap, and the like. Some may still be partially or fully green, depending upon their winter hardiness, but pruning is helpful to keep the plants full and under control. Tender perennials, like lantana and some of the ornamental grasses like pampas grass, should be left alone until mid-February. The dormant foliage can give the plant a little extra protection from the hardest freezes.
For The Birds – If your plants make seed that provide forage for birds, like coneflower and coreopsis, you may choose to leave them alone until the birds have eaten their fill (if not in a feature area). It will not harm the plants to do so.
Bananas and Elephant Ears – Go ahead and remove the mushy leftovers if the plants have not been cut back already. Mulch over the tops of the plants to keep them warm until spring.