Are climbing shoes supposed to hurt?

Climbing shoes are not meant to hurt, they are meant to be tight, which for some, do mean that they may hurt a little bit.

How tight they should be, though, depends on the climber.
Climbers need their climbing shoes to be tight so that they can stand on very tiny and delicate holds, and for their feet to no slip or move within the shoe, or at least to minimize that as much as possible.
This, though, is usually a requirement of high level climbers, those who have been climbing for years and can actually benefit from the added performance the shoes can give.

Beginner climbers on the other hand have no reason to wear shoes that are so tight, as the routes they will be climbing will not require such delicate and precise feet placements, and will not have such small holds.

As climbers progress up the grades and climb harder and harder routes, it does make sense to start getting tighter shoes, but only as required. There is no point in having super tight and expensive shoes if they don't actually bring you any value in comparison to beginner climbing shoes or to just looser climbing shoes, especially if the tighter shoes are just painful.

With all of that said, there is a slight level of pain that climbers may feel with climbing shoes, mainly with new shoes, but this has more to do with the fact that the shoes are just not comfortable, it shouldn't be actual pain.

Climbing shoes are not comfortable, and this discomfort may be the cause of some pain, but, this is temporary until your feet get used to climbing shoes.

Once you have been climbing for a while, you will actually start to feel quite comfortable in climbing shoes, and yes, you may even refer to them as "comfortable".

There is also a point to make about fitting the right shoe for the right foot.
Not all shoes fit everyone. So it is important to try on different pairs and to find the ones that actually fit your feet well. This is because climbing shoes are all made differently. They are made from different materials, and can have slightly different shapes and bends, especially in the toe area and the heel.