If a patient is not anesthetized, is it possible to remove their organs without harming the organs?

Anesthesia isn't really there to protect the organs, per say, but the comfort and well-being of the patient. Then anesthesia used in surgery is meant to sedate and paralyze the patient so that there is no rampant body movement during the procedure. Imagine how difficult it would be to operate on a patient writhing in pain. Many anesthetics also numb pain and may cause amnesia so that patients don't form long term memories of any discomfort during the procedure. Lastly, an anesthesiologist will always monitor the vital signs of the patient to insure that there are no signs of cardiovascular or respiratory distress. While it's theoretically possible to procure donor organs without anesthesia, it would be extremely difficult to do so safely and highly unethical to harm a patient in such a manner.

Usually the organs used for transplantation are commonly the kidneys,liver,heart,pancreas and cornea. It is theoretically possible to remove the kidneys liver ,pancreas under spinal or epidural anaesthesia but I believe general anaesthesia will be required!

Maybe I didn’t understand what you asked, but the only organs that are removed are from someone who has died. Of course that wouldn’t hurt them without anesthesia. However, if you mean removing the old, diseased organ for it to be replaced the answer is no. Going the other way, anesthesia will always be used.

Of course.

Anesthetics are only used to prevent the patient from feeling pain during the procedure, it is not a requirement for surgery.