I know that cenotes were used for religious purposes in Central America, and that in South America caves were sometimes used for mummy burials, but I don't know anything about the Native uses of caves north of Mexico. Did Native people use caves for anything such as shelter, storage, burials, or religious purposes?
While there are countless archaeological sites in North America with "Cave" in the name, most of these refer to rockshelters. I'm assuming your question refers to deep cavern dark zones. There are numerous examples of North American dark zone caverns used for mining, storage, and ritual and religious purposes. I'll focus on two well-documented examples in the Southeast and Southwest.
Early pioneers in Kentucky and Tennessee reported Native artifacts and mummified remains in deep caverns. These discoveries were frequent enough to create a local tourism business in the early 1800s. Prehistoric Native Americans of this region quarried selenite and satinspar crystals, gypsum and other sulfate minerals from these caves to be used for paint, plaster, medicine, psychoactive ritual drugs, and (for the crystals) as decorative and ritual objects. Based on artifacts recovered from Mammoth Cave, recent research by Crothers suggests that gypsum mining was strongly associated with male initiation rites in deep caves in the Early Woodland Period (ca. 1000-200 BC).
Burials in the Southeast are more common at cavern entrances or in rockshelters than in dark zones. Archaeologists have found at least two bodies deep in Kentucky's Mammoth Cave, both who died by accident. One was a young boy who died of hypothermia and the other a mid-40s man who was crushed by a rock as he was mining.
Switching to the Southwest, there are several examples of deep caves used for ritual and religious purposes. The modern Pueblos and their ancestors use deep caves as shrines, in some cases undergoing long pilgrimages to visit them. The connection between Ancestral Puebloan and Mesoamerican religion is far too complex for this post, but the ceremonial use of deep caves may indicate common religious connections between the two.
Feather Cave in Lincoln County, New Mexico is an excellent example of an Ancestral Puebloan cave shrine. The deepest part of the cave, Arrow Grotto, can only be reached by crawling through a narrow shaft about 20 meters long. Arrow Grotto was covered with pictographs and included hundreds of religious offerings.
There were miniature bows, a medium-sized bow, a large assemblage of miniature reed arrows, miniature bows with arrows attached, fragments of large reed arrows, a white shell bead, crook pahos (prayer sticks), and a wooden ball of the type used by modern Pueblos in ceremonial races. Two miniature arrows had been thrust into a crack in the wall.
Feather Cave dates to about 1000-1400 AD and its pictographs and religious offerings are consistent with historic and modern Pueblo religious rituals. Ceremonial Cave near El Paso, Texas is another example of a Southwestern shrine cave. This site of ca. 700-1400 AD had two burials (a young woman and toddler girl), and a wide variety of artifacts left as offerings.
Interestingly enough, the Southwest's most famous cave system, Carlsbad Caverns, has no significant archaeological evidence of prehistoric use of the dark zone. This may be because nothing has been discovered, although these caves have been extensively explored. If you visit the main cavern entrance today, you will see pictographs ca. 1000-2000 years old at the mouth of the cave, which was extensively inhabited until the Spanish Colonial period.
There are numerous other examples of dark zone caverns used for ritual purposes in North America. Caves with rock art include Mud Glyph Cave (Tennessee), Picture Cave (Missouri), and other sites in Alabama and Georgia. There are also shaft cave/sinkhole burials in Trans-Pecos Texas, although this is borderline dark zone/rockshelter. Also, though it's not North America, there are several Taino rock art sites in the Caribbean where petroglyphs are carved in cavern dark zones. One of these Taino sites dates into the Spanish Colonial period and includes Christian imagery.
Selected Sources:
George Crothers, P. Willey, and Patty Jo Watson, Cave archaeology and the NSS: 1941–2006. Journal of Cave and Karst Studies, 2007, v. 69, no. 1, p. 27–34.
George M. Crothers, Early Woodland Ritual Use of Caves in Eastern North America, American Antiquity, Vol. 77, No. 3 (July 2012), pp. 524-541
Florence Hawley Ellis and Laurens Hammack, The Inner Sanctum of Feather Cave, a Mogollon Sun and Earth Shrine Linking Mexico and the Southwest, American Antiquity, Vol. 33, No. 1 (Jan., 1968), pp. 25-44