Climbing shoes should not hurt but more of a discomfort. The fit will be snug with toes curled. It is important to fit snug without pain. If the shoe hurts, the final performance is lower as you will be more focused on the pain than what you are doing, and you can also have problems with blisters and calluses.
Even the stability is fundamental: if the feet hurt, your stability decreases and the climb is harder than it is, as you are more focused on the pain than on the climbing.
Keep also in mind that a pain in the foot can cause difficulties in managing the impact in case of a fall.
We hope we have helped you to buy the best climbing shoes for your needs!
Have a look at how to take care of your shoes here.