When the angular speed w of an object increases, its angular momentum mayor may not also increase. Give an example in which it does and one in which it does not.

Consider the motion of the Earth around the Sun, in the elliptical orbit. When the Earth gets closer to the Sun it speeds up which increases its angular velocity. However the angular momentum of the Earth remains constant. It is important to note that in this case, the increase in momentum of the Earth is compensated by distance between the Earth and the Sun. This maintains the constant angular momentum while increasing the angular velocity.
Therefore the increase in angular velocity of Earth is not followed by the increase in angular momentum.
Now consider a rotating wheel, which is given a torque by an external agent. The torque increases the angular velocity of the wheel which in turn increases the angular momentum.
Therefore the increase in angular velocity also increases the angular momentum.