Will pacifiers hurt my child's teeth?

Probably not. Depends on duration and intensity. Thumb or finger sucking much more damaging than pacifier. Sucking reflex is calming/soothing for infants. They should age out by 18-24 months. Continuing digit habit beyond age 4 can deform upper jaw and restrict lower jaw growth.
Maybe. It depends on how long the paci is used for and for how much each day. Generally speaking if pacis are used until about the age of one and then stopped, then there is no harm. But long term, consistent use clearly can cause shifting of teeth and disrupt normal bite patterns.
Not likely. Pacifier-sucking and thumbsucking generally do not damage teeth, but they will lead to more overbite in a baby's front teeth. Be sure these habits go away before the permanent teeth come. Throw out the pacifiers no later than when throwing out the bottles (around age 20-24 months). Since many or most kids get braces to straighten their teeth in their teen years, an overbite is correctable.