Most moves are not affected by happiness, but the move Return does more damage based on happiness. It has has a linear formula with 1 base power at no happiness and 102 base power at max happiness. Conversely, Frustration does less as a Pokemon is happier. The base power scale is the reverse of Return.
In order to Move Tutor various moves, such as Draco Meteor in Pokemon Sun and Moon, the Pokemon needs max happiness. This has occurred in previous games. For reference, moves which have required max happiness to tutor in ORAS include Blast Burn, Draco Meteor, Fire Pledge, Frenzy Plant, Grass Pledge, Hydro Cannon and Water Pledge.
Occasionally, there are specific in game items that depend on happiness. An example is the Footprint Ribbon given by Dr. Footprint for max Happiness in Gen IV games. I am currently unaware of any such items in Pokemon Sun and Moon.