Is it normal if my teeth hurt after a dental appointment?

Only short term: It depends on the treatment you had done. It is normal to have some post operative discomfort after most dental procedures, from the injection, extraction, gum irritation, abrasion, etc. But the purpose of dental procedures is to alleviate any discomfort or pain, so if the discomfort persists for more than say 7 days, i would go back to the dentist so he or she can assess what's wrong.

Not really: What type of procedure did you have? Shouldnt be sensitive for more than a day or two. It is possible that a very deep filling can have affected the nerve in such a way that a tooth will be very sensitive and you will need to get back to the treating dentist.

Maybe?: It depends on what you had done. A cleaning can cause short term sensitivity in the gums and awareness in the teeth to hot and cold. A new restoration may be sensitive to biting or temperature. If your symptoms persist longer than 3-4 days you should contact your dentist.