“Can general anesthesia for a tonsillectomy cause any damage?”

There is risk to any invasive procedure, but you need to trust your team.
Board certified anesthesiologist are medical doctors and have completed have postgraduate residency training that can take anywhere from 4 to 9 years depending on if they specialize. I hate to admit it but I couldn't do my job well if we did not have a bond. If you trust your surgeon, then trust your anesthesiologist, be as frank with them with your concern as I hope you are with your surgeon.

Yes, all anesthesia have side effects.