The motion will follow the path of the stronger force on the object.

Below is a top-view diagram of a box sitting on a floor. Students are asked to describe the path/motion of the box, assuming that the forces remain constant. 

The weight and the normal force will be equal and opposite (since the box remains on the floor), so they won't affect the box's motion. The box must be accelerating both to the right and toward the top of the page, since it experiences a net force with components in both of these directions. The box will travel to the right as well as toward the top of the page. Note, however, that since the rightward force is larger, it will speed up faster to the right than in the upward direction. So it will travel further rightward than upward.

Another way to think about this one, is to consider that the two forces shown are components of one force that points in the direction shown below. Thus we can see that the box will travel in a straight line along this direction, accelerating as it moves.