Before thinking about complicated quantum groups, it's best to first understand the simplest situation. So let's just look at when G is a finite group and A is the Hopf algebra of functionals on G. In the finite dimensional setting A-comodules is the same thing as $A^*$-modules, and in this case $A^*$ is just the group ring $\mathbb{C}[G]$. Modules over the group ring are exactly the same thing as group representations. This is the simplest example that you should always have in mind when thinking about quantum groups.
So in this specific case your question becomes "Are all finite dimensional unitary representations of finite groups self-dual?" The answer to that is obviously no. (If this answer is not obvious to you, then I'd strongly recommend reading an intro book on group representations before trying to tackle quantum group representations.)
Fancier examples would be to take A to be the ring of functions on SU(3) or to be the quantum group O_q(SU(3)). In either case the defining vector representation won't be self-dual.