A situation that describes a non-contact force changing the speed of an object is "a gust of wind blowing a leaf away." In this case, the wind (a non-contact force) exerts a force on the leaf, causing it to accelerate and change its speed.
The other options involve contact forces: the rider pedaling the bicycle, the ball rolling down a hill (which involves gravitational force, but the acceleration comes from the ball physically interacting with the hill), and the user pulling the handle of the drawer (a direct contact force).