I've been told to take the Band-Aid off a small cut every once in a while in order for it to heal faster. Is this good advice, or should I try to keep it covered as much as possible?
Presumably, you've been told to take the Band-Aid off periodically in order to give it air. In fact, this slows healing, so if you wish to have your cut heal as quickly as possible, keep it bandaged.
The primary benefit of a bandage is to help seal the wound and stop or slow bleeding and prevent re-injury and infection. Where people start to differ in opinion is once the wound has been cleaned and placed in a sterile environment. At this point, the primary benefit of bandages are to keep the wound moist, which your well-meaning friend might not realize provides optimal conditions for healing.
[dressings] are designed to create a moist wound healing environment which allows the wound fluids and growth factors to remain in contact with wound, thus promoting autolytic debridement and accelerating wound healing. 
Because of this, airing out the wound is counterproductive.
Exposing a wound to the air so it can breathe is a [mistake], because it creates a dry environment that promotes cell death. 
The moist environment has the bonus of sustaining a provisional extra-cellular matrix throughout the healing process, facilitating cell migration and ultimately, epithelialization. This results in less visible scarring once the wound is healed. 
 NYT Health