I'm going to have ten teeth out, does it hurt much with all the anesthesia?

No: Pulling out one tooth or several ones is the same under anesthesia . Post operative discomfort is the same but instead of being in one spot is going to be at several ones .. Which could be controlled with some pain killer ..

Shouldn't hurt: As long as you have adequate anesthesia, the only thing you should feel is just pressure of the teeth coming out. You should not feel any pain. If you feel pain, you need more anesthetic. If there is an infection in the area, this sometimes can reduce the effect of the local anesthetic. Discuss with dentist.