Many stories focus on sex and romance. Those are overwhelmingly the kinds of intimacy featured in fiction. Even outside the immediate sphere of erotic and romantic stories, they comprise major subplots in most genres and many stories. Attention to other types, expressions, and experiences of intimacy is rare. This largely ignores nonsexual family relationships such as siblings or parent/child. It shortchanges close professional relationships such as a cop's beat partner or a soldier's buddy. It tends to leave asexual people off the map altogether. Even for readers who like stories about sex and romance, this can get old -- especially if the writer doesn't pay any attention to the development of intimacy but just shoves the characters into bed as fast as possible.
Nonsexual forms of intimacy can add a great deal of depth and variety to fiction. On one end of the spectrum, they provide extra steps to support the journey from meeting a potential mate through romance, sex, and marriage. In the middle, they convey the import of family and professional connections, distinguishing those from more casual acquaintances. On the other end, they form much of the glue in primary relationships for people who don't base their ties on sexuality. Sex and romance are valuable, but they're not everything. Nonsexual intimacies are the "show don't tell" conveyance for the rest of the serious relationship field. Here are some examples and their story influence.
EDIT 6-26-17: See also "Five Types of Intimacy Other Than Sexual," which categorizes things somewhat differently than I do.
Personal & Body Care
This category covers stuff that has to do with body boundaries and maintenance. Ordinary adults do some of these things for themselves. Parents do some for their children. Caretakers or hired professionals may do them for people who can't manage on their own or just want someone else to do it. These things can express caring or comfort in a relationship, with varying degrees of intimacy.
Hair care. Brushing, braiding, washing, cutting -- all of these involve a lot of careful touching in ways that many people enjoy. Hair braiding is a bonding experience in some cultures. In fact, grooming is a bonding technique for social primates in general. People without close ties to others often treat themselves to regular salon visits as a socially acceptable way to meet the need for touch and interaction.
Shaving. This involves an unusually high level of trust, especially if the person is using a straight-edge razor or something else with an exposed blade rather than just a buzzer. Although it can apply to women, shaving is one of the few forms of physical intimacy that is most closely associated with men due to their facial hair. Initiaton into shaving is a major milestone for becoming a man, not just for boys during puberty but also for transsexuals during transition.
Bathing. This varies by culture; in America most people bathe alone but some other cultures practice communal bathing. A bath is usually more intimate than a shower, although a public bath can be non-intimate and small shower stall can be intimate. It's also different when two people wash each other (an exchange of intimacy and affection) than when one person washes someone else (more of a caretaking or protective gesture).
Feeding. A classic romantic motif involves lovers feeding each other, but it works as a way of providing and caring for someone in any context. Like bathing, it can also clue whether both parties are participating equally or one is taking care of the other (temporarily or regularly). This one has an existential flavor since survival depends on food supply.
Massage. The tone can be clinical, casual, nonsexually intimate, or erotic but it all comes down to a lot of skin contact. Some cultures, such as Swedish and Japanese, are far more comfortable with massage than American culture is; but you can still find it in America. Some Asian traditions offer orgasm (a "happy ending") as a non-erotic physical release, which is useful in contexts where erotic interaction is not desired but the body's needs are demanding.
Taking care of someone sick/injured. A natural part of family life, this can also crop up between professional partners or even strangers in some circumstances. It involves one person doing things that the other normally does alone, but currently finds difficult or impossible. This is a great way to break down walls for one of those stubborn characters who is impregnable under ordinary circumstances -- hence the popularity of hurt/comfort fiction.
Touching parts of the body not usually handled by strangers. The body divides into areas with different permissions. Strangers may shake hands, casual friends may slap each other on the shoulder. Only close relationships tend to involve touching the face, feet, inside forearms, nape of neck, etc.
Seeing someone without their adaptive equipment on. This includes glasses, dentalware, prosthetic limbs, a wheelchair, etc. Adaptive equipment is part of one's presentation to the everyday world, and taking it off can be as intimate as removing clothing, for many people in many contexts.
Removing or putting on someone's glasses. This one is worth special mention both because it's the most common version of a not-very-common motif, and because it's intimate without being overwhelming. It's something one might do for a friend who falls asleep on the couch, for instance. That makes it a good way to show that a relationship is becomingintimate.
Undressing someone. This can be kind of a one-way experience if the recipient isn't awake, and is often awkward for both people if they are awake. Sometimes it happens because hands are out of commission, but a more common example is someone passing out drunk. Overheating is another good reason. Different circumstances can imply different levels of intimacy.
(Read Part 2, Part 3, Part 4, Part 5.)