Winter dormancy for temperate plants is not unlike hibernation for some animals when cold weather and a shortening of the photo-period arrives gradually in autumn and winter. For temperate plants like Venus flytraps (Dionaea) and American Pitcher Plants (Sarracenia) and many sundews (Drosera) it is very important to respect this rest period, for without it a plant may get sickly and die.
During dormancy many of the pitchers, traps and leaves will turn brown or black and die back to base of the plant. You can gently remove these once they have died back completely. Below is a photo of a Venus flytrap in summer and then in winter:
Photo-period is the length of time the sun is out. If we take North Carolina as an example, where Venus flytraps are native, the photo-period at the summer solstice in June is roughly 15 hours of daylight. At the spring and autumn equinox the photo-period is about 12 hours. At the beginning of winter in December it shortens to about 9 hours, and then slowly, about a minute a day, it lengthens. For typical North American temperate plants, the crucial time for this cooling and short photo-period is about Thanksgiving (late November) to Valentines Day (mid February).
Plants outdoors in bog gardens that are set in the ground are much more protected than plants in exposed pots above ground. Bog gardens can offer protection for most plants even up, in the often frigid winter climates, around the Great Lakes and upper Midwest to New England. If early snow arrives before deeply freezing temperatures do, the snow will act as a wonderful insulator, keeping the bog just below freezing (32° F) as the snow cover keeps the icy air above the dormant plants. In autumn as the plants slow down and go dormant, many folks will trim the dying leaves away and cover the bog with extra protections such as a layer of hay, pine needles or burlap. This will greatly add to the bog's protection. When the worst of winter is ending the protective covering can be removed.
If you have potted plants outdoors they can easily take temperatures down into the 20s F for brief periods, but continuous cold below the upper teens can kill them. Therefore the plants should be moved to a more protected area, such as the coolest north facing windowsills (where direct sun won't warm the pots), or garage windowsills, or even basement window wells, where I'd recommend also covering them with a layer of mulch. You can also help by running refrigerated water through the pots once or twice a day, or adding ice cubes of purified water to the soil surface to slowly melt and add to the cooling, if the plants are kept indoors.
Some folks think the winter temperatures need to get very cold or similar to their native environment but this isn't true. It's really the short photo-period that instigates dormancy. At California Carnivores in Sonoma County, north of San Francisco, we frequently get outdoor temperatures in the 30s F and even 20s F and sometimes the teens, without any harm to temperate plants. The greenhouse however is heated to a "sweltering" 50° F if temperatures get below this outdoors, to protect tropical plants such as highland Nepenthes. Day temperatures in the greenhouse when the sun is out can warm to the 70s F without harm to the dormant plants. Our temperate plants still go dormant and rest happily but will begin growth earlier than the outdoor plants, sometimes as early as late January or February. No harm has ever been seen. So indoor areas such as lightly heated porches and windowsills in cool rooms can often be a good place for the plants. Remember to keep the soil of dormant plants damp to wet even during their rest period.
Refrigerating temperate plants for at least two months (December and January) is something I usually recommend only to growers in sub-tropical areas such as Miami or the lowlands of Hawaii. Usually someplace indoors can be found in regular homes in frigid winter climates. If you do refrigerate your plants, they may be bare-rooted in plastic zip lock bags with a handful of damp sphagnum peat moss or long fibered moss. Check them weekly for any fungus or mold, and if this occurs treat them with a dose of a fungicide such as Physan. This is also true for wintering windowsill plants that may develop mold on the soil. If temperatures are above freezing you can also place them outdoors in the rain which washes away mold spores and fungus. One drought year when winter rains were sparse, I saw in northern California bogs dormant buds of native sundews being attacked by botrytis fungus (a fuzzy gray fungus) because rain wasn't washing the fungus and spores away.
Allowing temperate plants a rest period from late autumn to late winter gives them vitality and correct flowering times the following year. It also give growers a break from the typical routines during the growing season, and when they begin to grow again in late winter and spring, it's a happy time and like seeing old friends again!