Are there any problems that require exponential time in quantum computing?
Most problems that require exponential time on a classical computer also require exponential time on a quantum computer.  For example EXPTIME-complete problems will require exponential time on either a classical or a quantum computer.

The class of problems solvable in polynomial time on a quantum computer is called BQP.  It is believed unlikely that BQP contains the NP-complete problems.  Thus the NP-complete problems probably take exponential time on quantum computers.

One of the most interesting problems that is known to be in BQP, but believed not to be in P is the integer factoring problem.  But integer factoring is believed unlikely to be NP-complete.