I usually don't add oils to my marinades, and they turn out just fine. I don't even alway include an acid (though usually do when dealing with cheaper, tougher cuts of meat). Although usually I add either an acid or a salt-based liquid (like soy sauce or viatnamese fish sauce), or both. I think perhaps technically salt-based liquids are considered brines rather than marinades.
I also often use what would probably be more of a rub than a marinade (a small amount of wet ingredients to dry ingredients, usually more of a paste though than a dry rub). For example, a bit of soy sauce, dried minced or powdered garlic or minced fresh or bottled garlic, seasoned salt or adobo seasoning, and maybe a squeeze of citrus juice. It's still pretty wet, but it's not enough for the meat to really soak in. Still, the flavors do penetrate the meat, especially since I often allow the meat to stay in the rub/marinade for several hours before cooking.
A lot depends on what you want the marinade to do. If you want it to actually tenderize as well as flavor the meat, you probably want an acid. Fat can also help in tenderizing, so you do have to watch how you cook meats if you don't add fat. You've got to be more careful not to overcook meats that have very little fat of their own and none from a marindade. However, if you're not needing to tenderize the meat, there's no reason not to use an oil-free rub or marinade.