Not really. They maybe uncomfortable and borderline painful, but you shouldn't actually feel any pain.
If you have blisters or you aren't used to climbing shoes, that could change the situation. Until your blisters go away or until you get used to your shoes, you could feel some pain.
Climbing shoes, after their break in period, should not be painful to walk in, but they can be uncomfortable, tight, and snug.
During a pair of climbing shoes break in period, they may be slightly painful, but should not cause too much pain. After their break in period, they should have stretched enough to not be painful, while still remaining tight and snug.
If you have been climbing in your shoes for a few sessions, or a few months, and they are still painful, you may have just not bought the right shoes for your feet, or they may just be too small.
Usually the break in period can be around 5 sessions, for some it may be more, but after a month or two or regular climbing, they should be broken in.
