Yes they do. Unfortunately, they don't like to sleep at the same time we do. Most mosquitoes are active at night or at dusk and dawn, and rest or sleep during the day.
They look for sheltered places, such as brush or thick weeds, caves or rock shelters, holes in the ground, hollow logs or holes in trees. When they live close to people, they use the handy structures we've built, hiding in basements, barns, culverts under roads, in closets or any dark place where they expect to be undisturbed. They don't sleep lying down, though. They rest in about the same position as when they land on you.
Scientists who wanted to study mosquitoes discovered it was easier to catch them when they were asleep or resting, by building wooden "resting boxes" to attract them. From that work, we know that mosquitoes like shady, secluded places to sleep, rather than bright, open areas. Few chose to sleep in resting boxes placed in fields or yards, but they would congregate in boxes in shady forests, as long as there weren't too many other choices, since they liked to rest amid the dense undergrowth of the forest as much as in the man-made boxes nearby. They were more apt to seek out shelter facing west, so they were shaded from the morning sun.
Mosquitoes who bite at night start seeking places to sleep as soon as daylight comes. Entomologist Wayne J. Crans reported, "We found mosquito populations build steadily within the boxes during the morning hours and remain static until midafternoon." The mosquitoes he found, though, were usually ones who bite only at night. Some mosquitoes bite in the day as well as the night, and so they sleep at different times. Aedes aegypti mosquitoes will bite in the morning or late afternoon, so they're more apt to rest in the middle part of the day.  They prefer to rest indoors, in closets, on hanging clothes or towels, or other dark secluded locations. 
A few mosquitoes, like the Asian Tiger mosquito, bite during the day, so they may be more likely to sleep at night, and they prefer to rest outdoors rather than inside.
In winter, mosquitoes which hibernate sleep even more deeply. They look for sheltered locations to shield them from the coldest temperatures, such as caves, basements, overhanging banks or among tree roots, and go into a deeper dormancy until warm weather comes. Entomologist J. Turner Brakeley discovered they assume a position he called the "hibernation squat."  He observed that they bend their legs and tuck their body close to the surface they rest on, probably to get as much warmth from it as possible. Since they're cold-blooded, they can't make their own body heat, but they can absorb it from stone or earth which may be closer to cave temperature than air temperature.
Sometimes even hibernating mosquitoes will wake up on a warm winter day and emerge from hiding, then go back to sleep when cold weather returns until spring. The genera Anopheles, Culex, and Culiseta have many species that sleep through the winter in the United States, but not all kinds of mosquitoes hibernate. Many lay eggs in the fall, then the adults die, and the next generation comes from the eggs that hatch in the spring.
 John B. Silver, Mosquito Ecology: Field Sampling Methods
 Scott C. Weaver, Frontiers in Dengue Virus Research
 John B. Smith, Report of the New Jersey State Agricultural Experiment Station
Public domain photos from the CDC.