My DM says that my cleric has to drop his morningstar
to cast spells. Is he right?

Yes and no. To cast a spell with a somatic (S) component,
you must gesture freely with at least one hand. (Player’s
Handbook, page 140) A cleric (or any caster, for that matter)
who holds a weapon in one hand and wears a heavy shield on
the other arm doesn’t have a hand free to cast a spell with a
somatic component (which includes most spells in the game).
To cast such a spell, the character must either drop or sheathe
his weapon.

Another simple option is for the cleric to carry a buckler or
light shield instead of a heavy shield. The buckler leaves one
hand free for spellcasting, and you don’t even lose the
buckler’s shield bonus to AC when casting with that hand. The
light shield doesn’t give you a free hand for spellcasting, but
since you can hold an item in the same hand that holds the light
shield, you could switch your weapon to that hand to free up a
hand for spellcasting. (You can’t use the weapon while it’s held
in the same hand as your shield, of course.) The rules don’t
state what type of action is required to switch hands on a
weapon, but it seems reasonable to assume that it’s the
equivalent of drawing a weapon (a move action that doesn’t
provoke attacks of opportunity).