Interesting Question. Anesthesia works by suppressing neurotransmitters and decreasing the oxygen levels in the brain.
When you are "under" from anesthesia your mind is not really recording or sensing the same way you normally do.
As for the aspect of those reporting having left their bodies while under anesthesia, you get similar stories from those who passed out from smoke inhalation, drowning and trauma that rendered them unconscious.
Whether this has a lasting effect on the soul, who can truly say.
The affect on your soul is based on the stress and troubles you carry with you and weigh your soul with these burdens.
Anesthesia has no long term effects, and has been proven to help recovery and take your mind off the treatment. It can increase the recovery time by slowing down the healing process, but affecting your soul in negligible if none at all.
Cast off your fears, leave the stones where you find them; then the weight you carry on your journey will be a lighter one.